Spin to photon transducer

ABSTRACT

Methods, devices, and systems are described for storing and transferring quantum information. An example device may comprise at least one semiconducting layer, one or more conducting layers configured to define at least two quantum states in the at least one semiconducting layer and confine an electron in or more of the at least two quantum states, and a magnetic field source configured to generate an inhomogeneous magnetic field. The inhomogeneous magnetic field may cause a first coupling of an electric charge state of the electron and a spin state of the electron. The device may comprise a resonator configured to confine a photon. An electric-dipole interaction may cause a second coupling of an electric charge state of the electron to an electric field of the photon.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a divisional application of U.S. applicationSer. No. 16/534,431, “Spin To Photon Transducer,” filed on Aug. 7, 2019,which claims the benefit of and is a non-provisional of U.S. ApplicationNo. 62/715,533, “Spin To Photon Transducer,” filed on Aug. 7, 2018, eachof which is incorporated herein by reference in its entirety for any andall purposes.

GOVERNMENT RIGHTS

This invention was made with government support under Grant No.W911NF-15-1-0149 awarded by the U.S. Army/Army Research Office and GrantNo. H98230-15-R-0453 awarded by the U.S. Department of Defense. Thegovernment has certain rights in the invention.

BACKGROUND

Electron spins in silicon quantum dots are attractive systems forquantum computing owing to their long coherence times and the promise ofrapid scaling of the number of dots in a system using semiconductorfabrication techniques. Nearest-neighbor exchange coupling of two spinshas been demonstrated but has great limitations due to the inherentlimitations in scale. Spin-photon coupling has been studied but ischallenging because of the small magnetic-dipole moment of a singlespin, which limits magnetic-dipole coupling rates to less than 1kilohertz. Thus, there is a need for more sophisticated techniques forestablishing quantum information systems.

SUMMARY

The present disclosure describes methods, systems, and devices forstoring and transferring quantum information. An example device maycomprise at least one semiconducting layer, one or more conductinglayers configured to define at least two quantum states in the at leastone semiconducting layer and confine an electron in or more of the atleast two quantum states, and a magnetic field source configured togenerate an inhomogeneous magnetic field. The inhomogeneous magneticfield may cause a first coupling of an electric charge state of theelectron and a spin state of the electron. The device may comprise aresonator configured to confine a photon. An electric-dipole interactionmay cause a second coupling of an electric charge state of the electronto an electric field of the photon.

An example system may comprise a first structure configured to define atleast two first quantum states in a first semiconducting layer andconfine a first electron in or more of the at least two first quantumstates. The first structure may comprise at least one magnetic fieldsource configured to supply an inhomogeneous magnetic field to the firstelectron. The system may comprise a second structure configured todefine at least two second quantum states in a second semiconductinglayer and confine a second electron in or more of the at least twosecond quantum states. The system may comprise a resonator disposedadjacent the first structure and the second structure. Tuning of anexternal magnetic field may allow for (e.g., may configure) a photon inthe resonator to mediate coupling a first spin state of the firstelectron to a second spin state of the second electron.

An example method may comprise confining a first electron with a firststructure. The first structure may define two quantum states in at leastone semiconducting layer using one or more conducting layers. The methodmay comprise causing, based on an inhomogeneous magnetic field source, afirst coupling of an electric charge state of the first electron and afirst spin state of the first electron. The method may comprise causing,based on an electric-dipole interaction, a second coupling of anelectric field of a photon in a resonator to an orbital state of thefirst electron. The method may comprise causing, based on the firstcoupling and the second coupling, a coupling of the photon and the firstspin state of the first electron.

This Summary is provided to introduce a selection of concepts in asimplified form that are further described below in the DetailedDescription. This Summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used to limit the scope of the claimed subject matter. Furthermore,the claimed subject matter is not limited to limitations that solve anyor all disadvantages noted in any part of this disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

The file of this patent or application contains at least onedrawing/photograph executed in color. Copies of this patent or patentapplication publication with color drawing(s)/photograph(s) will beprovided by the Office upon request and payment of the necessary fee.

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate embodiments and together with thedescription, serve to explain the principles of the methods and systems.

FIG. 1A shows a block diagram of an example system for transferringquantum state information.

FIG. 1B shows a diagram of an example structure for transferring quantumstate information.

FIG. 2A shows an optical image of the superconducting microwave cavity.

FIG. 2B shows a false-colour scanning electron micrograph (SEM) of aDQD.

FIG. 2C shows a schematic cross-sectional view of the DQD device.

FIG. 2D shows cavity transmission amplitude A/A₀ at f=f_(c).

FIG. 2E shows A/A₀ as a function of ε with V_(B2)=710 mV and a fit tocavity input-output theory (black dashed line), with g_(c)/(2π)=40 MHz.

FIG. 2F shows 2t_(c)/h as a function of V_(B2) for DQD1, obtained bymeasuring A(ε)/A₀ at different values of V_(B2).

FIG. 3A shows A/A₀ as a function of the cavity drive frequency f and anexternally applied magnetic field B_(z) ^(ext) for DQD1.

FIG. 3B shows A/A₀ as a function of ƒ for DQD1 at B_(z) ^(ext)=90.3 mTand B_(z) ^(ext)=92.2mT.

FIG. 3C shows A/A₀ as a function of ƒ for DQD2 at B_(z) ^(ext)=91.1 mTand B_(z) ^(ext)=92.6 mT.

FIG. 4A shows A/A₀ as a function of ƒ and B_(z) ^(ext) at ε=0 (left),ε=20 μeV (about 4.8 GHz; middle) and ε=40 μeV (about 9.7 GHz; right),with 2t_(c)/h=7.4 GHz.

FIG. 4B shows spin-photon coupling rate g_(s)/(2π) (top) and spindecoherence rate γ_(s)/(2π) (bottom) as functions of 2t_(c)/h, with ε=0(data).

FIG. 4C shows DQD energy levels as a function of ε.

FIG. 5A shows cavity phase response Δϕ at f=f_(c).

FIG. 5B shows electron spin resonance (ESR).

FIG. 5C shows schematic showing the experimental sequence for coherentspin control and measurement.

FIG. 5D shows Δϕ as a function of τ_(B), with 2t_(c)/h=11.1 GHz andε′=70 μeV, showing single-spin Rabi oscillations.

FIG. 6 shows micromagnet design.

FIG. 7 shows photon number calibration.

FIG. 8A shows a DQD stability diagram.

FIG. 8B shows another DQD stability diagram.

FIG. 8C show another DQD stability diagram.

FIG. 8D show another DQD stability diagram.

FIG. 9A shows an example spin decoherence rate.

FIG. 9B shows another example spin decoherence rate.

FIG. 9C shows another example spin decoherence rate.

FIG. 9D shows another example spin decoherence rate.

FIG. 10A shows vacuum Rabi splittings for 2t_(c)/h<f_(c).

FIG. 10B shows vacuum Rabi splittings for 2t_(c)/h>f_(c).

FIG. 11 shows spin relaxation at ε=0.

FIG. 12A show theoretical fits to vacuum Rabi splittings.

FIG. 12B show theoretical fits to vacuum Rabi splittings.

FIG. 13A shows the ratio 2g_(s)/(κ/2+γ_(s)) as a function of 2t_(c)/h.

FIG. 13B shows the ratio g_(s)/γ_(s) as a function of 2t_(c)/h.

FIG. 14 shows a schematic illustration of the Si gate-defined DQDinfluenced by an homogeneous external magnetic field.

FIG. 15A shows energy levels as a function of the DQD detuningparameter.

FIG. 15B shows energy levels as a function of the DQD detuningparameter.

FIG. 15C shows a schematic representation of the ∧ system that capturesthe essential dynamics in FIG. 15A.

FIG. 15D shows a schematic representation of the ∧ system that capturesthe essential dynamics in FIG. 15B.

FIG. 16A shows expected effective coupling g_(s)/g_(c)−|d₀₁₍₂₎|

FIG. 16B shows spin-photon coupling strength g_(s)/g_(c) and spindecoherence rate γ_(s)/γ_(c) as a function of t_(c) for ϵ=0, B_(z)=B_(z)^(res), and B_(x)=1.62 μeV.

FIG. 17 shows cavity transmission spectrum, |A|, as a function of B_(z)and ϵ at zero driving frequency detuning Δ₀=0.

FIG. 18A shows cavity transmission spectrum |A| as a function of B_(z)and Δ₀.

FIG. 18B shows |A| as a function of B_(z) (Δ₀) for the value of Δ₀(B_(z)) indicated by the dashed line.

FIG. 18C shows |A| as a function of B_(z) (Δ₀) for the value of Δ₀(B_(z)) indicated by the dashed line.

FIG. 19 shows cavity transmission |A| as a function of Δ₀ close to theresonant field for different values of the charge-cavity couplingg_(c)/2π={40,80,160} MHz.

FIG. 20 shows strength of an example spin-cavity coupling.

FIG. 21A shows expected spin-photon coupling strength as a function ofthe DQD detuning and the tunnel coupling.

FIG. 21B shows cavity transmission |A| as a function of Δ₀.

FIG. 22A shows an example optical micrograph of the superconductingcavity containing two single electron DQDs.

FIG. 22B shows an example false-color scanning electron microscope imageof the L-DQD.

FIG. 22C shows cavity transmission A/A₀ as a function of B^(ext) and ƒ.

FIG. 22D shows that increasing the field angle to ϕ=2.8° dramaticallyreduces the field separation.

FIG. 23A shows expected spin resonance frequencies as a function of ϕfor B^(ext)=106.3 mT (top panel) and B^(ext)=110 mT (bottom panel).

FIG. 23B shows A/A₀ as a function of ƒ and ϕ demonstrates simultaneoustuning of both spins into resonance with the cavity at ϕ=5.6° andB^(ext)=106.3 mT.

FIG. 23C shows an example theoretical prediction for A/A₀.

FIG. 24A shows tuning the L-spin into resonance with the cavity resultsin vacuum Rabi splitting with magnitude 2g_(s,L).

FIG. 24B shows A/A₀ as a function of ƒ and B^(ext) with ϕ=5.6° indicatesan enhanced vacuum Rabi splitting when the L-spin and R-spin are tunedinto resonance with the cavity.

FIG. 24C shows A/A₀ as a function of ƒ for the R-spin in resonance(upper curve), L-spin in resonance (middle curve), and both spins inresonance (bottom curve).

FIG. 24D shows cavity-assisted spin-spin coupling is also observed in asecond device with a different gate pattern.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The present disclosure relates to a quantum computing device, methods,an systems. An example device may comprise a spin to photon transducer.The device may be configured to transfer the spin quantum state of asolid-state electron to a microwave photon in a superconducting cavity.

The disclosed device may comprise a hybrid nano-electronic deviceconfigured to convert the quantum state of a spin electron into a photonin a superconducting cavity. The device may convert quantum informationencoded in a single photon to spin state information. This conversionmay comprise the following steps. The electric field of the cavityphoton may be coupled to the orbital state of the electron throughelectric dipole interaction. The orbital state of the electron may behybridized with its spin state through a micro-magnet placed over thedouble quantum dot in which the electron resides. A large spin-cavitycoupling rate of 10 MHz may be achieved by the device, which satisfiesthe requirement for strong coupling and allows for scaling up individualspin qubits to form a full spin quantum processor via photonic links.

One application of the disclosed approach is to allow distant electronspins to interact through microwave photons. The disclosed approach mayallow electron spins to be coupled at long distances using a singlemicrowave frequency photon. The superconducting cavity used in thedevice is ˜7 mm long whereas the electron is confined within a spatialextent of ˜70 nm. By converting the electron spin into a microwavephoton which is free to move within the cavity, the device enablesdistant spins to be coupled through the photonic link. This is animportant for building a quantum computer using electron spins.

The disclosed approach is advantageous over previous devices whichcouple electron spins to microwave photons through magneticdipole-dipole interactions. Due to the weak magnetic field of a singlemicrowave photon, the dipole-dipole coupling strength is very weak(ranging from a few Hz to a few kHz), and limits the device operationspeed. The disclosed device couples microwave photons to electron spinsby hybridizing the electron spin with its orbital state, which respondsto the electric field of a single photon. This indirect coupling schemeallows a large spin-cavity coupling strength of over 10 MHz to beachieved, which exceeds the threshold necessary for transferring thequantum state of the electron spin to that of a microwave photon.

The disclosed device may comprise a Si/SiGe semiconductorheterostructure which confines electrons in a 2-dimensional plane. Threeoverlapping layers of aluminum electrodes may be patterned. Theseelectrodes may be biased to confine a single electron within a doublequantum dot (DQD). A niobium superconducting cavity may be fabricatedadjacent (e.g., next to, in contact with) to the DQD. The electric fieldof a cavity photon may be coupled to the orbital state of the electronthrough electric dipole interaction. A micro-magnet placed over the DQDmay produce a strong local magnetic field gradient which hybridizes theorbital state of the electron with its spin state, thereby coupling theelectric field of the cavity photon to the spin state of the electron.

The disclosed device successfully achieved the strong coupling regimebetween the electron spin and a cavity photon for the first time withgate-defined quantum dots. This strong coupling indicates that thedisclosed device may be used for long-range entanglement of distantspins. Further improvement of the device includes suppression of chargenoise, which may be the limiting factor of the coherence time of theelectron spin.

The disclosed device has been made and tested experimentally. VacuumRabi splitting, which is a splitting of the cavity transmission into twowell-separated peaks, is clearly observed when the transition frequencyof the electron spin is tuned close to the cavity frequency. Theobservation of vacuum Rabi splitting is well-established evidence forstrong coupling.

The disclosed approach may be used to rapidly read out spin qubits,which are a popular choice for building a solid-state quantum computer.The readout speed is only limited by the rate at which photons leave thecavity, which may be easily controlled. The cavity may also serve as abus which interconnects distant spin qubits together, which may beimportant for scaling up spin qubits into a full-fledged quantumcomputer.

FIG. 1A shows a block diagram of an example system for transferringquantum state information. The system 100 may comprise a first structure102 and/or a second structure 104. The first structure 102 may becoupled to the second structure 104 via a resonator 106 (e.g., or apath, waveguide, cavity, optical path). The first structure 102, thesecond structure 104, and/or the resonator 106 may be disposed on a chip108 (e.g., monolithically integrated circuit, substrate). The chip 108may comprise a first material stack comprising the first structure 102.The chip 108 may comprise a second material stack comprising the secondstructure 104. The second material stack may be separate from the firstmaterial stack. The first material stack and second material stack maybe formed simultaneously using the same fabrication process or may beformed as two separate processes. In some implementations, the firststructure 102 and the second structure 104 may be disposed on separatechips.

The first structure 102 may be configured to define at least two firstquantum states. The at least two first quantum states may comprise,have, and/or be indicative of spin states (e.g., electron spin states).The first structure 102 may define the at least two first quantum statesbased on one or more potential wells (e.g., generated using gates todefine electric fields). The at least two first quantum states maycomprise a gate defined silicon double quantum dot.

The at least two first quantum states may be defined in a firstsemiconducting layer. The first structure 102 may be configured toconfine a first electron in or more of the at least two first quantumstates. The first structure 102 may comprise at least one first magneticfield source (e.g., the first magnetic field source 134 and the secondmagnetic field source 136 shown in FIG. 1B). The first magnetic fieldsource may be configured to supply a first inhomogeneous magnetic fieldto the first electron.

The at least one first magnetic field source may comprise at least onemicro-magnet disposed in a material stack comprising the firstsemiconducting layer. The at least one first magnetic field source maycomprise a first micro-magnet and a second micro-magnet separated fromthe first micro-magnet. The at least one first magnetic field source maybe tilted such that a long axis of the at least one first magnetic fieldsource is angled relative to an axis between the at least two firstquantum states of the first structure 102. The at least one firstmagnetic field source may comprise a current carrying wire configured togenerate the first inhomogeneous magnetic field (e.g., to enablespin-photon coupling). The at least one first magnetic field source maycomprise an electrically tunable material, a multiferroic material, anelectrically tunable multiferroic material, a combination thereof,and/or the like.

The second structure 104 may be configured to define at least two secondquantum states in a second semiconducting layer and confine a secondelectron in or more of the at least two second quantum states. The atleast two second quantum states may comprise, have, and/or be indicativeof spin states (e.g., electron spin states). The second structure 104may define the at least two second quantum states based on one or morepotential wells (e.g., generated using gates to define electric fields).The at least two second quantum states may comprise a gate definedsilicon double quantum dot.

The at least two second quantum states may be defined in a secondsemiconducting layer. The second structure 104 may be configured toconfine a second electron in or more of the at least two second quantumstates. The second structure 104 may comprise at least one secondmagnetic field source. The at least one second magnetic field source maybe configured to supply a second inhomogeneous magnetic field to thesecond electron. The second electron may be separated from the firstelectron by a distance in a range of one or more of: about 1 mm to about5 mm, about 2 mm to about 4 mm, about 3 mm to about 4 mm, about 3 mm toabout 4 mm, about 1 mm to about 10 mm, or about 3 mm to about 8 mm.

The at least one second magnetic field source may comprise at least onemicro-magnet disposed in a material stack (e.g., the second materialstack) comprising the second semiconducting layer. The at least onesecond magnetic field source may comprise a third micro-magnet and afourth micro-magnet separated from the third micro-magnet. The at leastone second magnetic field source may be tilted such that a long axis ofthe at least one second magnetic field source is angled relative to anaxis between the at least two second quantum states of the secondstructure 104. The at least one second magnetic field source maycomprise a current carrying wire configured to generate the secondinhomogeneous magnetic field (e.g., to enable spin-photon coupling). Theat least one second magnetic field source may comprise an electricallytunable material, a multiferroic material, an electrically tunablemultiferroic material, a combination thereof, and/or the like.

The resonator 106 may be disposed adjacent the first structure 102and/or the second structure 104. The resonator 106 may comprise one ormore of a cavity, a superconductive cavity, or a microwave cavity. Theresonator 108 may comprise a path 110 (e.g., optical path), a firstmirror 112 disposed on the path 110, and/or a second mirror 114 disposedon the path 110 opposite the first mirror 112. The path may be definedby walls, a waveguide, a combination thereof, and/or the like.

The resonator 106 may comprise a center pin (e.g., or center wire)adjacent a vacuum gap. The center pin may be disposed along a centeraxis of the resonator 106. The center pin may have a thickness in arange of one or more of: about 50 nm to about 10 μm. or about 0.5 μm toabout 1 μm. The vacuum gap may have a thickness in a range of one ormore of: about 15 μm to about 25 μm or about 50 nm to about 30 μm. Theresonator 106 may have an impedance in a range of one or more of: about50Ω to about 200Ω, about 50Ω to about 300Ω, about 50Ω to about 3 kΩ,about 200Ω to about 300Ω, about 200Ω to about 2 kΩ, about 1 kΩ to about2 kΩ, about 200Ω to about 3 kΩ or about 1 kΩ to about 3 kΩ. A length ofthe resonator 106 may be between about 10,000 and about 1,000,000 timesa length in which the first electron is confined. The resonator 106 mayhave a center frequency in a range of one or more of: about 2 GHz toabout 12 GHz or about 7 GHz and about 8 GHz.

The system 100 may be configured to enable spin-photon coupling betweenthe first electron and a photon in the resonator 106. The spin-photoncoupling may be based on an external magnetic field. The system 100 maybe configured to enable spin-spin coupling between the first electronand the second electron. Tuning of the external magnetic field may allowfor the photon in the resonator 106 to mediate coupling a first spinstate of the first electron to a second spin state of the secondelectron.

The first inhomogeneous magnetic field may enable a first coupling of anelectric charge state of the first electron and a first spin state ofthe first electron. The coupling may comprise hybridization of theelectric charge state of the first electron and the first spin state ofthe first electron. An electric-dipole interaction may cause a secondcoupling of an electric charge state of the first electron to anelectric field of the photon. The second coupling may comprisehybridization of the electric charge state of the first electron and theelectric field of the photon.

The second inhomogeneous magnetic field may enable a third coupling ofan electric charge state of the second electron and a second spin stateof the second electron. The coupling may comprise hybridization of theelectric charge state of the second electron and the second spin stateof the second electron. An electric-dipole interaction may cause afourth coupling of an electric charge state of the second electron to anelectric field of the photon. The fourth coupling may comprisehybridization of the electric charge state of the second electron andthe electric field of the photon.

The photon may be coupled to the first spin state of the first electronbased on the first coupling and the second coupling. The photon may becoupled to the second spin state of the second electron based on thethird coupling and the fourth coupling. Adjusting one or more of a fieldstrength or an angle of an external magnetic field applied to one ormore of the resonator 106 or the first structure 102 may cause couplingof the first spin state to the photon. Adjusting one or more of thefield strength or the angle of an external magnetic field applied to oneor more of the resonator 106 or the second structure 104 may causecoupling of the second spin state to the photon. Causing resonance of anenergy of the first electron and an energy of the photon may causecoupling of the first spin state to the photon.

Causing resonance of an energy of the second electron and an energy ofthe photon may cause coupling of the second spin state to the photon.Adjusting one or more of an angle or an amplitude of the externalmagnetic field may allow for simultaneous tuning of the first spin stateof the first electron and the second spin state of the second electronin resonance with one or more of the photon or the resonator 106.Adjusting one or more of an angle or an amplitude of the externalmagnetic field may allow for simultaneous tuning of the a plurality ofspin states in resonance with one or more of the photon or the resonator106. A spin-photon coupling between the photon and one or more of thefirst electron or the second electron may occur with a spin-photoncoupling rate in a range of one or more of: about 10 Mhz to about 15 Mhzor about 5 MHz to about 150 MHz.

The system 100 may be configured to enable coupling of a plurality ofquantum states to the resonator 106 to generate long range quantum gatesbetween the plurality of quantum states. The plurality of quantum statesmay be coupled to and/or comprised in the resonator 106. A plurality ofstructures (e.g., in one or in separate material stacks) may each beconfigured to define one or more quantum states (e.g., potential well,energy states, spin states). The plurality of quantum states may each betuned (e.g., simultaneously, in groups, individually) in and/or out ofresonance with the resonator 106. The plurality of quantum states mayeach be tuned in and/or out of resonance with the resonator 106 usingelectrical control, electrical tuning, electrical detuning, tuning of amagnetic field (e.g., a local magnetic field associated with a specificstructure and/or quantum state), a combination thereof, and/or the like.Current carrying wires, multiferroic materials, a combination thereof,and/or the like may be used to tune corresponding quantum states of theplurality of quantum states.

FIG. 1B shows a diagram of an example structure 116 (e.g., or device),such as a quantum structure. The quantum structure may be configured tostore quantum state information. The first structure 102 and/or thesecond structure 104 of FIG. 1A may be implemented as the examplestructure 116.

The structure 116 may comprise one or more semiconducting layers, suchas first semiconducting layer 118, a second semiconducting layer 120, athird semiconducting layer 122, a fourth semiconducting layer 124, or acombination thereof. The one or more semiconducting layers may compriseone or more of silicon, germanium, or silicon germanium. The one or moresemiconducting layers may comprise an isotopically enriched material.The isotopically enriched material may comprise isotopically enrichedsilicon. The isotopically enriched silicon may comprise nuclear spinzero silicon 28 isotope. The first semiconducting layer 118 and thethird semiconducting layer 122 may comprise the same material, such assilicon or isotopically enriched silicon. The second semiconductinglayer 120 and the fourth semiconducting layer 124 may comprise the samematerial, such as a silicon germanium.

The structure 116 may comprise one or more conducting layers. The one ormore conducting layers may comprise a first conducting layer comprisingone or more barrier gates 126. The one or more barrier gates 126 may beconfigured to define at least two quantum states. The one or morebarrier gates may be configured to define one or more quantum potentialwells, such as a double quantum dot (e.g., gate defined double quantumdot), triple quantum dot, and/or the like.

The one or more conducting layers may comprise a second conductinglayer. The second conducting layer may comprise one or more plungergates 128 configured to cause an electron to move between the at leasttwo quantum states. The electron may be caused to move from one quantumstate to another quantum state based on applying a positive voltage toone of the one or more plunger gates 128 and/or apply a negative voltageto another of the one or more plunger gates 128. The second conductinglayer may comprise a source 130 and a drain 132. The source 130 and/orthe drain 132 may be configured to provide the electron for isolation inthe at least two quantum states.

The structure 116 may be coupled (e.g., electrically coupled) to aresonator, such as the resonator 106 of FIG. 1A. The one or moreconducting layers comprises a layer electrically coupled to theresonator. The layer coupled to the resonator may comprise the secondconducting layer. A plunger gate of the one or more plunger gates 128may be coupled to the resonator. The plunger gate may be coupled to apin (e.g., center pin, wire, center wire) of the resonator.

The layer electrically coupled to the resonator may comprise a thirdconducting layer. The third conducting layer may comprise split-gatelayer (e.g., as shown in FIG. 22B). The third conducting layer (e.g.,split gate layer) may comprise a first gate and a second gate separatedfrom the first gate by a gap. One of more of the first gate or thesecond gate may be coupled to the resonator. One of more of the firstgate or the second gate may be coupled to the resonator via a pin (e.g.,center pin, wire, center wire) of the resonator. One or more of a sizeor a location of the gap may increase (e.g., maximize, optimize) and/ormay be selected to increase an electric field in a region of the atleast two first quantum states. One or more of a size or a location ofthe gap may increase (e.g., maximize, optimizes) and/or may be electedto increase coherent coupling between one or more of the at least twofirst quantum states and the photon. The gap may be below or above oneor more of the one or more plunger gates or the one or more barriergates.

The structure 116 may comprise at least magnetic field source, such as afirst magnetic field source 134 and a second magnetic field source 136.The at least magnetic field source may comprise one or more magneticand/or one or more current carrying wires. An example magnetic materialmay comprise cobalt, iron, nickel, alloys thereof, a multiferroicmaterial, or a combination thereof. Any other magnetic material may beused. The at least magnetic field source may provide an inhomogeneousmagnetic field (e.g., a magnetic field gradient), and/or the like.

The system 100 of FIG. 1A and the structure 116 of FIG. 1B may bemanufactured using a variety of techniques described further herein. Insome implementations, the one or more conducting layers (e.g., or gatesthereof, such as the one or more barrier gates 126, the one or moreplunger gates 128) may be shorted together (e.g., temporarily) duringfabrication. The one or more conducting layers may be shorted togetherusing shorting straps. In a later stage of fabrication (e.g., or aftercompletion of fabrication), the shorting straps and/or shorting may beremoved and/or otherwise separated prevent later shorting between theone or more conducting layers.

Further description and examples of the disclosed techniques aredescribed in detail below in the following sections. It should beunderstood that any of the features of one section may be combined withany of the features of another section as an implementation of thedisclosure.

First Section: A Coherent Spin-Photon Interface in Silicon

A strong coupling between a single spin in silicon and a singlemicrowave-frequency photon is disclosed herein, with spin-photoncoupling rates of more than 10 megahertz. The mechanism that enables thecoherent spin-photon interactions may be based on spin-chargehybridization in the presence of a magnetic-field gradient. In additionto spin-photon coupling, coherent control and dispersive readout of asingle spin is demonstrated. These results open up a direct path toentangling single spins using microwave-frequency photons.

Solid-state electron spins and nuclear spins are quantum mechanicalsystems that can be almost completely isolated from environmental noise.As a result, they have coherence times as long as hours and so are oneof the most promising types of quantum bit (qubit) for constructing aquantum processor. On the other hand, this degree of isolation posesdifficulties for the spin-spin interactions that are needed to implementtwo-qubit gates. So far, most approaches have focused on achievingspin-spin coupling through the exchange interaction or the much weakerdipole-dipole interaction. Among existing classes of spin qubits,electron spins in gate-defined silicon quantum dots have the advantagesof scalability due to mature fabrication technologies and low dephasingrates due to isotopic purification. Currently, silicon quantum dots arecapable of supporting fault-tolerant control fidelities for single-qubitgates and high-fidelity two-qubit gates based on exchange. Coupling ofspins over long distances has been pursued through the physicaldisplacement of electrons and through ‘super-exchange’ via anintermediate quantum dot. The recent demonstration of strong couplingbetween the charge state of a quantum-dot electron and a single photonhas raised the prospect of strong spin-photon coupling, which couldenable photon-mediated long-distance spin entanglement. Spin-photoncoupling may be achieved by coherently hybridizing spin qubits withphotons trapped inside microwave cavities, in a manner similar to cavityquantum electrodynamics with atomic systems and circuit quantumelectrodynamics with solid-state qubits. Such an approach, however, isextremely challenging: the small magnetic moment of a single spin leadsto magnetic-dipole coupling rates of 10-150 Hz, which are far too slowcompared with electron-spin dephasing rates to enable a coherentspin-photon interface.

Here, this outstanding challenge is resolved by using spin-chargehybridization techniques disclosed herein to couple the electric fieldof a single photon to a single spin in silicon. Spin-photon couplingrates g_(s)/(2π) of up to 11 MHz were measured, nearly five orders ofmagnitude higher than typical magnetic-dipole coupling rates. Thesevalues of g_(s)/(2π) exceed both the photon decay rate κ/(2π) and thespin decoherence rate γ_(s)/(2π), firmly anchoring our spin-photonsystem in the strong-coupling regime.

The disclosed coupling scheme may comprise (e.g., or consist of) twostages of quantum-state hybridization. First, a single electron may betrapped within a gate-defined silicon double quantum dot (DQD) that hasa large electric-dipole moment. A single photon confined within amicrowave cavity may hybridize with the electron charge state throughthe electric-dipole interaction. Second, a micrometre-scale magnet(micromagnet) placed over (e.g., over in a material stack, or adjacent)the DQD may hybridize electron charge and spin by producing aninhomogeneous magnetic field. The combination of the electric-dipoleinteraction and spin-charge hybridization may give rise to a largeeffective spin-photon coupling rate. At the same time, the relativelylow level of charge noise in the device may ensure that the effectivespin decoherence rate γs remains below the coherent coupling rate gs—acriterion that has hampered previous efforts to achieve strongspin-photon coupling.

As well as demonstrating a coherent spin-photon interface, the discloseddevice architecture may be configured single-spin control and readout.Single-spin rotations may be electrically driven. The resulting spinstate may be detected through a dispersive phase shift in the cavitytransmission, which reveals Rabi oscillations.

Spin-photon interface. An example device according to the presentdisclosure that enables strong spin-photon coupling is shown in FIG. 2A.The device may comprise two gate-defined DQDs fabricated using anoverlapping aluminium gate stack, as shown in FIG. 2B. The gates may beelectrically biased to create a double-well potential that confines asingle electron in the underlying natural-silicon quantum well, asillustrated in FIG. 2C. A plunger gate (P2) on each DQD may be connectedto a centre pin of a half-wavelength niobium superconducting cavity witha centre frequency of f_(c)=5.846 GHz and quality factor of Q_(c)=4,700(κ/(2π)=f_(c)/Q_(c)=1.3 MHz), which may hybridize the electron chargestate with a single cavity photon through the electric-dipoleinteraction. Because the spin-photon coupling rate g_(s) is directlyproportional to the charge-photon coupling rate g_(c) (refs 25, 31-34,39-41), the cavity dimensions (FIG. 2A, inset) have been modified toachieve a high characteristic impedance Zr and therefore a highg_(c)(g_(c)∝√{square root over (Z_(r))}). To hybridize the charge stateof the trapped electron with its spin state, a cobalt micromagnet may befabricated near (e.g., adjacent, in the same material stack) the DQD,which may generate an inhomogeneous magnetic field. For our devicegeometry, the magnetic field due to the cobalt micromagnet has acomponent along the z axis B_(z) ^(M) that is approximately constant forthe DQD and a component along the x axis that takes on an average valueof B_(x,L) ^(M) (B_(x,R) ^(M)) for the left (right) dot (e.g., as shownin FIG. 2C and FIG. 6 ). The relatively large field difference B_(x,R)^(M)−B_(x,L) ^(M)=2B_(x) ^(M) leads to spin-charge hybridization, which,when combined with charge-photon coupling, gives rise to spin-photoncoupling.

The strength of the charge-photon interaction may be characterizedfirst, because this sets the scale of the spin-photon interaction rate.For simplicity, only one DQD is active at a time for all of themeasurements presented here. The cavity is driven by a coherentmicrowave tone at frequency f=f_(t) and power P≈−133 dBm (correspondingto approximately 0.6 photons in the cavity, determined on the basis ofAC Stark shift measurements of the spin-qubit frequency in thedispersive regime; see e.g., FIG. 7 ). The normalized cavitytransmission amplitude A/A₀ is displayed in FIG. 2D as a function of thevoltages V_(P1) and V_(P2) on gates P1 and P2 of the first DQD (DQD1),which reveals the location of the (1, 0)↔(0, 1) inter-dot chargetransition (e.g., FIGS. 8A-D show overall stability diagrams). Here (n,m) denotes a charge state, with the number of electrons in the left (P1)and right (P2) dot being n and m, respectively. The charge-photoncoupling rate is estimated quantitatively by measuring A/A₀ as afunction of the DQD level detuning ε (FIG. 2E). By fitting the data withthe cavity input-output theory model using κ/(2π)=1.3 MHz, g_(c)/(2π)=40MHz and 2t_(c)/h=4.9 GHz are determined, where t_(c) is the inter-dottunnel coupling and h is the Planck constant. A charge decoherence rateof γ_(c)/(2π)=35 MHz is also estimated from the fit and confirmedindependently using microwave spectroscopy with 2t_(c)/h=5.4 GHz (refs19, 20, 42). Fine control of the DQD tunnel coupling, which is importantfor achieving spin-charge hybridization, is shown in FIG. 2F, in which2t_(c)/h is plotted as a function of the voltage V_(B2) on the inter-dotbarrier gate B2. A similar characterization of the second DQD (DQD2)yields g_(c)/(2π)=37 MHz and γ_(c)/(2π)=45 MHz at the (1, 0)↔(0, 1)inter-dot charge transition. Owing to the higher impedance of theresonator, the values of g_(c) measured here are much larger than inprevious silicon DQD devices, which is helpful for achieving strongspin-photon coupling. In general, there are device-to-device variationsin γ_(c) (refs 19, 43). It is unlikely the slightly higher chargedecoherence rate is a result of our cavity design, because the Purcelldecay rate is estimated to be Γ_(c)/(2π)≈0.02 MHz«γ_(c)/(2π). Excitedvalley states are not visible in the cavity response of either DQD,suggesting that they have negligible population. Therefore valleys areexcluded from the analysis below.

Strong single spin-photon coupling. Strong coupling is demonstratedbetween a single electron spin and a single photon, as evidenced by theobservation of vacuum Rabi splitting. Vacuum Rabi splitting occurs whenthe transition frequency of a two-level atom f_(a) is brought intoresonance with a cavity photon of frequency f_(c) (refs 21, 23).Light-matter hybridization leads to two vacuum-Rabi-split peaks in thecavity transmission. For our single-spin qubit, the transition frequencybetween two Zeeman-split spin states is f_(a)≈E_(z)/h, whereE_(z)=gμ_(B)B_(tot) is the Zeeman energy and the approximate sign is dueto spin-charge hybridization, which shifts the qubit frequency slightly.Here g is the g-factor of the electron, μ_(B) is the Bohr magneton andB_(tot)=√{square root over ([(B_(x,L) ^(M)+B_(x,R) ^(M))/2]²+(B_(z)^(M)+B_(z) ^(ext))²)} is the total magnetic field. To bring f_(a) intoresonance with f_(c), the external magnetic field B_(z) ^(ext) is variedalong the z axis while measuring the cavity transmission spectrum A/A₀as a function of the drive frequency f (FIG. 3A). Vacuum Rabi splittingsare clearly observed at B_(z) ^(ext)=−91.2 mT and B_(z) ^(ext)=92.2 mT,indicating that E_(z)/h=f_(c) at these field values and that the singlespin is coherently hybridized with a single cavity photon. Thesemeasurements are performed on DQD1, with 2tc/h=7.4 GHz and ε=0. Thedependence of g_(s) on ε and t_(c) is investigated below. Assuming g=2for silicon, it is estimated that an intrinsic field of about 120 mT isadded by the micromagnet, comparable to values found in a previousexperiment using a similar cobalt micromagnet design.

To further verify the strong spin-photon coupling, cavity transmissionspectrum at B_(z) ^(ext)=92.2 mT is plotted, as shown in FIG. 3B. Thetwo normal-mode peaks are separated by the vacuum Rabi frequency2g_(s)/(2π)=11.0 MHz, giving an effective spin-photon coupling rate ofg_(s)/(2π)=5.5 MHz. The photon decay rate at finite magnetic field isextracted from the line width of A/A₀ at B_(z) ^(ext)=90.3 mT, at whichE_(z)/h is largely detuned from f_(c), yielding κ/(2π)=1.8 MHz. A spindecoherence rate of γs/(2π)=2.4 MHz, with contributions from both chargedecoherence and magnetic noise from the Si nuclei, is extracted frommicrowave spectroscopy in the dispersive regime with 2tc/h=7.4 GHz andε=0 (FIGS. 9A-D), confirming that the strong-coupling regimeg_(s)>γ_(s), κ has been reached. The spin-photon coupling rate obtainedhere is more than four orders of magnitude larger than rates currentlyachievable using direct magnetic-dipole coupling to lumped-elementsuperconducting resonators.

The local magnetic field that is generated using cobalt micromagnets isvery reproducible, as evidenced by examining the other DQD in thecavity. Measurements on DQD2 show vacuum Rabi splittings at B_(z)^(ext)=±92.6 mT (FIG. 3A, insets). The spin-photon coupling rate andspin decoherence rate are determined to be g_(s)/(2π)=5.3 MHz andγs/(2π)=2.4 MHz, respectively as shown in FIG. 3C. These results arehighly consistent with DQD1, and so henceforth this discussion willfocus on DQD1.

Electrical control of spin-photon coupling. For quantum informationapplications it is desirable to turn qubit-cavity coupling rapidly onfor quantum-state transfer and rapidly off for qubit-state preparation.Rapid control of the coupling rate is often accomplished by quicklymodifying the qubit-cavity detuning f_(a)-f_(c). Practically, suchtuning can be achieved by varying the qubit transition frequency f_(a)with voltage or flux pulses or by using a tunable cavity. Theseapproaches are not directly applicable for control of the spin-photoncoupling rate because f_(a) depends primarily on magnetic fields thatare difficult to vary on nanosecond timescales. In this section, it isshown that control of the spin-photon coupling rate can be achievedelectrically by tuning ε and t_(c) (refs 32, 40).

First the ε dependence of gs is investigated. In FIG. 4A measurementsare shown of A/A₀ as a function of B_(z) ^(ext) and f for ε=0, ε=20 μeVand ε=40 μeV. At ε=20 μeV (about 4.8 GHz), vacuum Rabi splitting isobserved at B_(z) ^(ext)=92.1 mT with a spin-photon coupling rate ofg_(s)/(2π)=1.0 MHz that is substantially lower than the value ofg_(s)/(2π)=5.5 MHz obtained at ε=0. At ε=40 μeV (about 9.7 GHz), only asmall dispersive shift is observed in the cavity transmission spectrumat B_(z) ^(ext)=91.8 mT, suggesting a further decrease in gs. Theseobservations are qualitatively understood by considering that at ε=0 theelectron is delocalized across the DQD and forms molecular bonding (|−

) or anti-bonding (|+

) charge states (FIG. 4C). In this regime, the cavity electric fieldleads to a displacement of the electron wavefunction of the order of 1nm (Methods). Consequently, the electron spin experiences a largeoscillating magnetic field, resulting in a substantial spin-photoncoupling rate. By contrast, with |ε|»t_(c) the electron is localizedwithin one dot and it is natural to work with a basis of localizedelectronic wavefunctions |L

and |R

, where L and R correspond to the electron being in the left and rightdot, respectively (e.g., as shown in FIG. 4C). In this effectivelysingle-dot regime, the displacement of the electron wavefunction by thecavity electric field is estimated to be about 3 pm for a single-dotorbital energy of E_(orb)=2.5 meV, greatly suppressing the spin-photoncoupling mechanism. The large difference in the effective displacementlengths between the single-dot and double-dot regimes also implies animprovement in the spin-photon coupling rate at ϵ=0 of approximately twoorders of magnitude compared to |ε|»t_(c). Alternatively, the reductionof g_(s) may be interpreted as a result of suppressed hybridizationbetween the |−, ↑

and |+, ↓

states due to their growing energy difference at larger |ε|, as evidentfrom FIG. 4C (see discussion below). Here ↑ (↓) denotes an electron spinthat is aligned (anti-aligned) with B_(z) ^(ext). These measurementshighlight the important role of charge hybridization in the DQD.

Additional electric control of g_(s) is enabled by voltage tuning t_(c)(e.g., see FIG. 2F). FIG. 4B shows g_(s)/(2π) and γ_(s)/(2π) asfunctions of 2t_(c)/h at ε=0, as extracted from vacuum Rabi splittingmeasurements and microwave spectroscopy of the electron spin resonance(ESR) transition line width (e.g., see FIGS. 3B, 5B, 9A-C, 10A-B). Bothrates increase rapidly as 2t_(c)/h approaches the Larmor precessionfrequency E_(z)/h≈5.8 GHz, and a spin-photon coupling rate as high asg_(s)/(2π)=11.0 MHz is found at 2t_(c)/h=5.2 GHz. These trends areconsistent with the DQD energy-level spectrum shown in FIG. 4C. With2t_(c)/h»E_(z)/h and ε=0, the two lowest energy levels are |−, ↓

and |−, ↑

and the electric-dipole coupling to the cavity field is small. As 2t_(c)is reduced and made comparable to E_(z), the ground state remains |−, ↓

but the excited state becomes an admixture of |−, ↑

and |+, ↓

owing to the magnetic-field gradient B_(x,R) ^(M)−B_(x,L) ^(M)=2B_(x)^(M) and the small energy difference between the states. The quantumtransition that is close to resonance with E_(z) is now partiallycomposed of a change in charge state from − to +, which respondsstrongly to the cavity electric field and gives rise to larger values ofg_(s). For 2t_(c)/h<E_(z)/h, a decrease in t_(c) increases the energydifference between |−, ↑

and |+, ↓

, which reduces their hybridization and results in a smaller g_(s). Itshould be noted that hybridization with charge states increases thesusceptibility of the spin to charge noise and relaxation, resulting inan effective spin decoherence rate γ_(s) that is also strongly dependenton t_(c) (e.g., see FIG. 4B). Theoretical predictions of g_(s) and γ_(s)as functions of 2t_(c)/h, based on measured values of g_(c) and γ_(c)(e.g., see FIG. 2E) are in good agreement with the data (e.g., see FIG.4B). The discrepancy in the fit of γ_(s) is discussed in Methods. Theelectric control of spin-photon coupling demonstrated here allows thespin qubit to switch quickly between regimes with strong coupling to thecavity and idle regimes in which the spin-photon coupling rate andsusceptibility to charge decoherence are small.

Dispersive readout of a single spin. The preceding measurementsdemonstrate the ability to couple a single electron spin to a singlephoton coherently, potentially enabling long range spin-spin couplings.For the device to serve as a building block for a quantum processor, itmay also be necessary to prepare, control and read out the spin state ofthe trapped electron deterministically. First spin transitions areinduced by driving gate P1 with a continuous microwave tone of frequencyf_(s) and power P_(s)=−106 dBm. When f_(s)≈E_(z)/h, the excited-statepopulation of the spin qubit P_(↑) increases and the groundstate-population P_(↓) decreases. In the dispersive regime, in which thequbit-cavity detuning Δ/(2π)≈E_(z)/h−f_(c) satisfies Δ/(2π)»g_(s)/(2π),the cavity transmission experiences a phase response Δϕ≈ tan⁻¹[2g_(s)²/(κΔ)] for a fully saturated (P_(↑)=0.5) qubit. It is thereforepossible to measure the spin state of a single electron by probing thecavity transmission. As a demonstration, the ESR transition isspectroscopically probed by measuring Δϕ as a function of ƒ_(s) andB_(z) ^(ext) (e.g., see FIG. 5A). These data are acquired with2t_(c)/h=9.5 GHz and ε=0. The ESR transition is clearly visible as anarrow feature with Δϕ≠0 that shifts to higher f_(s) with increasingB_(z) ^(ext). Δϕ also changes sign as B_(z) ^(ext) increases, consistentwith the sign change of the qubit-cavity detuning Δ when the Larmorprecession frequency E_(z)/h exceeds f_(c). The nonlinear response inthe small region around B_(z) ^(ext)=92 mT is due to the breakdown ofthe dispersive condition |Δ/(2π)»g_(s)/(2π).

Cohereing single-spin control and dispersive spin-state readout isdemonstrated. For these measurements, ε=0 and 2t_(c)/h=11.1 GHz werechosen to minimize the spin decoherence rate γ₅ (e.g., FIG. 4B). Herethe spin-photon coupling rate g_(s)/(2π)=1.4 MHz (e.g., FIG. 4B). Theexternal field is fixed at B_(z) ^(ext)=92.18 mT, which ensures that thesystem is in the dispersive regime with Δ/(2π)=14 MHz»g_(s)/(2π). Ameasurement of Δϕ(f_(s)) in the low-power limit (e.g., FIG. 5B) yields aLorentzian line shape with a full-width at half-maximum of 0.81 MHz,which corresponds to a low spin decoherence rate of γ_(s)/(2π)=0.41 MHz(refs 19, 42). Qubit control and measurement are achieved using thepulse sequence illustrated in FIG. 5C. Starting with a spin-down state|↓

at ε=0, the DQD is pulsed to a large detuning ε′=70 μeV (about 17 GHz),which decouples the spin from the cavity. A microwave burst withfrequency f_(s)=5.874 GHz, power P_(s)=−76 dBm and duration τ_(B) issubsequently applied to P1 to drive a spin rotation. The DQD is thenpulsed adiabatically back to ε=0 for a fixed measurement time T_(M) fordispersive readout. To reinitialize the qubit, T_(M)=20 μe»T₁(ε=0) ischosen, where T₁(ε=0)=3.2 μe is the spin relaxation time measured at ε=0(e.g., FIG. 11 ). FIG. 5D displays the time-averaged Δϕ as a function ofτ_(R), obtained with an integration time of 100 ms for each data point.Coherent single-spin Rabioscillations were observed with a Rabifrequency of f_(R)=6 MHz. In contrast to readout approaches that rely onspin-dependent tunneling, the disclosed dispersive cavity-based readoutcorresponds in principle to quantum nondemolition readout. The readoutscheme is also distinct from previous work that used a cavity-coupledInAs DQD, which detects the spin state through Pauli blockade ratherthan spin-photon coupling. In addition to enabling single spin-photoncoupling, the disclosed device may be configured for preparing,controlling and dispersively reading out single spins.

A coherent spin-photon interface was realized at which a single spin ina silicon DQD is strongly coupled to a microwave-frequency photonthrough the combined effects of the electric-dipole interaction andspin-charge hybridization (e.g., see Methods for a discussion of theprospects of applying the spin-photon interface to realize cavitymediated spin-spin coupling). Spin-photon coupling rates of up to 11 MHzare measured in the device, exceeding magnetic-dipole coupling rates bynearly five orders of magnitude. The spin decoherence rate is stronglydependent on the inter-dot tunnel coupling t_(c) and ranges from 0.4 MHzto 6 MHz, possibly limited by a combination of charge noise, chargerelaxation and remnant nuclear field fluctuations. All-electric controlof spin-photon coupling and coherent manipulation of the spin state aredemonstrated, along with dispersive readout of the single spin, whichlays the foundation for quantum non-demolition readout of spin qubits.These results could enable the construction of an ultra-coherent spinquantum computer with photonic interconnects and readout channels, withthe capacity for surface codes, ‘all-to-all’ connectivity and easyintegration with other solid-state quantum systems such assuperconducting.

Methods:

Device fabrication and measurement. The Si/SiGe heterostructurecomprises (e.g., or consists of) a 4-nm-thick Si cap, a 50-nm-thickSi_(0.7)Ge_(0.3) spacer layer, a 8-nm-thick natural-Si quantum well anda 225-nm-thick Si_(0.7)Ge_(0.3) layer on top of a linearly gradedSi_(1-x)Ge_(x) relaxed buffer substrate. Design and fabrication detailsfor the superconducting cavity and DQDs are described elsewhere. Theapproximately 200-nm-thick Co micromagnet is defined using electron beamlithography and lift off. In contrast to earlier devices, the gatefilter for P1 was changed to an L₁-C-L₂ filter, with L₁=4 nH, C=1 pF andL₂=12 nH. This three-segment filter allows microwave signals below 2.5GHz to pass with less than 3 dB of attenuation.

All data are acquired in a dilution refrigerator with a base temperatureof 10 mK and electron temperature of T_(e)=60 mK. The measurements ofthe transmission amplitude and phase response of the cavity (e.g., FIGS.2A-F, 5A-D) are performed using a homodynedetection scheme. Themeasurements of the transmission spectra of the cavity (e.g., FIGS.3A-C, 4A-C) are performed using a network analyser. The microwave driveapplied to P1 (e.g., FIGS. 5A-D) is provided by a vector microwavesource and the detuning pulses are generated by an arbitrary waveformgenerator, which also controls the timing of the microwave burst (e.g.,FIG. 5D).

To maximize the magnetization of the Co micromagnet and minimizehysteresis, data at positive (negative) external applied magnetic fields(e.g., FIG. 3A) are collected after B_(z) ^(ext) is first ramped to alarge value of +300 mT (−300 mT). A small degree of hysteresis stillremains for the micromagnet of DQD1, as can be seen by the differentmagnitudes of B_(z) ^(ext) at which positive- and negative-field vacuumRabisplittings are observed (e.g., FIG. 3A). In FIG. 5A, the slope ofthe ESR transition is d(E_(z)/h)/dB_(z) ^(ext)=44 MHz mT⁻¹, which ishigher than the value (28 MHz mT⁻¹) expected for a fully saturatedmicromagnet. The slope of the transition suggests that the micromagnetis not fully magnetized and has a magnetic susceptibility of dB_(z)^(M)/dB_(z) ^(ext)≈0.6 around B_(z) ^(ext)=92 mT.

Estimate of displacement length. The displacement length of the electronwavefunction by the cavity electric field may be estimated byconsidering the spin-photon coupling strength. For g_(s)/(2π)=10 MHz,the effective AC magnetic field B_(ac) ^(ESR) that drives ESR is B_(ac)^(ESR)=[g_(s)/(2π)][h/(gμ_(B))]≈0.4 mT. The field gradient for our DQDis 2B_(x) ^(M)/l≈0.3 mT nm⁻¹, where l=100 nm is the inter-dot distance.Therefore, the effective displacement of the electron wavefunction isestimated to be about 1 nm in the DQD regime. For a single dot, thespin-photon coupling strength is expected to be g_(s)/(2π)≈(gμ_(B)B_(x)^(M)/E_(orb))(g_(c)/2π)≈30 kHz (refs 31, 33) for an orbital energy ofE_(orb)=2.5 meV. The equivalent AC magnetic field that is induced by thecavity is therefore B_(ac) ^(ESR)≈1 μT, corresponding to a displacementlength of only about 3 pm.

Conversion of cavity phase response to spin population. For thedispersive readout of the Rabi oscillations (e.g., FIG. 5D), thetheoretically expected cavity phase response is ϕ_(↑)=tan⁻¹[2g_(s)²/(κΔ)]=9.6° when the spin qubit is in the excited state, andϕ_(↑)=tan⁻¹[2g_(s) ²/(κΔ)]=−9.6° when the spin qubit is in the groundstate. Because our measurement is averaged over T_(M)»T₁, spinrelaxation during readout will reduce the phase contrast observed in theexperiment. To enable a conversion between the phase response of thecavity Δϕ and the excited-state population of the spin qubit P_(↑), thespin relaxation time T₁ is measured by fixing the microwave burst timeat τ_(B)=80 ns, which corresponds to a π pulse on the spin qubit. Thephase response of the cavity Δϕ is then measured as a function of T_(M)for T_(M)>5 μs>T₁ (FIG. 11 ). The result is fitted to a function of theform Δφ=ϕ₀+ϕ₁(T₁/T_(M))[1−exp (−T_(M)/T₁)] to extract T₁=3.4 μs, whereϕ₀ and ϕ₁ are additional fitting parameters. The effects of the cavityringdown time 1/κ≈90 ns and the π-pulse time of 80 ns are ignored in thefit, because both of these times are much shorter than the measurementtime T_(M). The phase contrast that results from the fit, ϕ₁≈17.7°, isclose to the maximum contrast expected at this spin-photon detuning,ϕ_(↑)−v_(↓)=19.2°. On the basis of this value of T₁, the measured phaseresponse is converted into the excited-state population viaP_(↑)=(1+Δϕϕ_(↑,r))/2, where ϕ_(↑,r)=ϕ_(↑)(T₁/T_(M))[1−exp(−T_(M)/T₁)]=1.5° is the reduced phase response due to spin relaxationduring the readout time T_(M)=20 μs. The converted spin population P_(↑)shown in FIG. 5D has a visibility of about 70%, which could be improvedby performing single-shot measurements.

Input-output theory for cavity transmission. Here is briefly summarizedthe theoretical methods used to calculate the cavity transmission A/A₀shown in FIG. 2E and FIG. 12A-B. Starting from the Hamiltonian thatdescribes the DQDH ₀=½(ετ_(z)+2t _(c)τ_(x) +B _(z) σ+B _(x) ^(M)σ_(x)τ_(Z)where τ_(x) and τ_(z) are Pauli operators that act on the orbital chargestates of the DQD electron, σ_(x) and σ_(z) are Pauli operators that acton the spin states of the electron, B_(z)=B_(z) ^(ext)=B_(z) ^(M)denotes the total magnetic field along the z axis and B_(x)^(M)=(B_(x,R) ^(M)−B_(x,L) ^(M))/2 is half the magnetic field differenceof the DQD in the x direction. In the theoretical model, the assumptionis made that the average magnetic field in the x direction satisfiesB_(x) ^(M)=(B_(x,R) ^(M)+B_(x,L) ^(M))/2=0, which is a goodapproximation given the geometry of the micromagnet and its alignmentwith the DQD. The electric-dipole coupling is added to the cavity withthe HamiltonianH ₁ =g _(c)(a+a ^(†))T _(z)where a and a^(†) are the photon operators for the cavity. Theelectric-dipole operator can be expressed in the eigenbasis {|n

} of H₀ as

$\tau_{z} = {\sum\limits_{n,{m = 0}}^{3}{d_{nm}\left. ❘n \right\rangle\left\langle m❘ \right.}}$

The quantum Langevin equations for the operators a and σ_(nm)=|n

m|:

${\overset{.}{a} = {{{i\Delta_{0}a} - {\frac{k}{2}a} + {\sqrt{k_{1}}a_{{in},1}}} = {{\sqrt{k_{2}}a_{{in},2}} - {{ig}_{c}e^{i\omega_{R}t}{\sum\limits_{n,{m = 0}}^{3}{d_{nm}\sigma_{nm}}}}}}}{{\overset{.}{\sigma}}_{nm} = {{{- {i\left( {E_{m} - E_{n}} \right)}}\sigma_{nm}} - {\sum\limits_{n^{\prime}m^{\prime}}{\gamma_{{nm},n^{\prime}}\sigma_{n^{\prime}m^{\prime}}}} + {\sqrt{2_{\gamma}}F} - {{{ig}_{c}\left( {{ae}^{{- i}\omega_{R}t} + {a^{\dagger}e^{i\omega_{R}t}}} \right)}d_{{mnP}_{nm}}}}}$where Δ₀=ω_(R)−ω_(c) is the detuning of the driving field frequency(ω_(R)=2πf) relative to the cavity frequency (ω_(c)=2πf_(c)) andP_(nm)=P_(n)−P_(m) is the population difference between levels n and m(P_(n) can, for example, be assumed to be a Boltzmann distribution inthermal equilibrium). This description is equivalent to a more generalmaster-equation approach in the weak-driving regime in which populationchanges in the DQD can be neglected. Furthermore, κ₁ and κ₂ are thephoton decay rates at ports 1 and 2 of the cavity and α_(in,1) is theinput field of the cavity, which is assumed to couple through port 1only (a_(in,2)=0). The quantum noise of the DQD F is neglected in whatfollows. The super-operator γ with matrix elements γ_(nm,n′m′)representsdecoherence processes, including charge relaxation and dephasing due tocharge noise (these processes also imply some degree of spin relaxationand dephasing due to spin-charge hybridization via B_(x) ^(M)). The goalis to relate the incoming parts a_(in,j) of the external field at theports to the outgoing fields a_(out,i)=√{square root over(κ_(i)a)}−a_(in,i). The transmission A=ā_(out,2)/ā^(τ) _(in,1) (wherethe overbars denote time-averaged expectation values) through themicrowave cavity is then computed using a rotating-wave approximation toeliminate the explicit time dependence in last equation above, bysolving the equations for the expected value of these operators in thestationary limit (ā and σ _(n,m)):

$A = {- \frac{i\sqrt{\kappa_{1}\kappa_{2}}}{{- \Delta_{0}} - {i{\kappa/2}} + {g_{c}{\sum\limits_{n = 0}^{2}{\sum\limits_{j = 1}^{3 - n}d_{n,{n + {j\chi_{n,{n + j}}}}}}}}}}$where χ_(n,n+j)=σ _(n,n+j)/ā are the single-electron partialsusceptibilities and d_(ij) are the dipole-transition matrix elementsbetween DQD states.

Theoretical models for spin-photon coupling and spin decoherence. Hereis presented a brief derivation of the analytical expressions for thespin-photon coupling rate g_(s) and the spin decoherence rate γ_(s). Thefocus is on the ε=0 regime used in FIG. 4B. Accounting for spin-chargehybridization due to the field gradient B_(x) ^(M), the relevanteigenstates of the DQD Hamiltonian in the equation above are |0

≈|−, ↑

,|1

=cos(ϕ/2)|−, ↑

+sin(ϕ/2)|+, ↓

=sin(ϕ/2)|−, ↑

−cos(ϕ/2)|+, ↓

and |3≈|+, ↑

. Here is introduced a mixing angle ϕ=tan⁻¹[gμ_(B)B_(x)^(M)/(2t_(c)−gμ_(B)B_(z))]. The dipole-transition matrix element for theprimarily spin-like transition between |0

and |1

is d₀₁≈ sin(ϕ/2) and the dipole-transition matrix element for theprimarily charge-like transition between |0

and |2

is d₀₂≈ cos(ϕ/2). The transition between |0

and |3

is too high in energy (off-resonance) and is therefore excluded from ourmodel. The spin-photon coupling rate isg_(s)=g_(c)|d₀₁|=g_(c)|sin(ϕ/2)|, in agreement with previous theoreticalresults.

To calculate the effective spin decoherence rate γ_(s) ^(c) that arisesfrom charge decoherence, first constructed are the operators τ₀₁=|0

1|≈ cos(ϕ/2)σ_(s)+sin(ϕ/2)σ_(r) and σ₀₂=|0

|≈ sin(ϕ/2)σ_(s)−(ϕ/2)σ_(s)−cos(ϕ/2)σ_(r). Here σ_(s)=|−, ↓

−, ↑| and σ_(r)=|−, ↓

|

+, ↓| are lowering operators for the electron spin and charge,respectively. Assuming that the electron charge states have a constantdecoherence rate γ_(c)=γ₁/2+γ_(ϕ), where γ₁ is the charge relaxationrate and γ_(ϕ) is a dephasing rate due to charge noise, the equations ofmotion for these operators are

${{\overset{.}{\sigma}}_{01} = {\gamma_{c}\left\lbrack {{{- {\sin^{2}\left( \frac{\Phi}{2} \right)}}\sigma_{01}} + {\frac{\sin(\Phi)}{2}\sigma_{02}}} \right\rbrack}}{{\overset{.}{\sigma}}_{02} = {\gamma_{c}\left\lbrack {{\frac{\sin(\Phi)}{2}\sigma_{01}} - {{\cos^{2}\left( \frac{\Phi}{2} \right)}\sigma_{02}}} \right\rbrack}}$

Combined with charge-photon coupling, the overall equations of motion ina rotating frame with a drive frequency f≈f_(c) assume the form

${\overset{.}{a} = {{i\Delta_{0}a} - {\frac{K}{2}a} + {\sqrt{\kappa_{1}}a_{{in},1}} - {{ig}_{c}\left( {{d_{01}\sigma_{01}} + {d_{02}\sigma_{02}}} \right)}}}{\overset{.}{\sigma_{01}} = {{{{- i}\delta_{1}\sigma_{01}} - {\gamma_{c}{\sin^{2}\left( \frac{\Phi}{2} \right)}\sigma_{01}}} = {{\gamma_{c}\frac{\sin(\Phi)}{2}\sigma_{02}} - {{ig}_{c}{ad}_{10}}}}}{{\overset{.}{\sigma}}_{02} = {{{- i}\delta_{2}\sigma_{02}} - {\gamma_{c}{\cos^{2}\left( \frac{\Phi}{2} \right)}\sigma_{02}} + {\gamma_{c}\frac{\sin(\Phi)}{2}\sigma_{01}} - {{ig}_{c}{ad}_{20}}}}$

The δ₁ and δ₂ terms are defined as δ₁/(2π)=(E₁−E₀)/h−f andδ₂/(2π)=(E₂−E₀)/h−f, where E_(0,1,2) correspond to the energy of the |0

, |1

and |2

state, respectively. Steady-state solutions to the above equations givethe electric susceptibility of the spin qubit transitionX_(0,1)=σ₀₁/a=g_(s)/(δ₁−iγ_(s) ^((c))), where a charge-induced spindecoherence rate is identified as γ_(s) ^((c))=γ_(c)[δ₂ sin²(ϕ/2)+δ₁cos²(ϕ/2)]/δ₂. To account for spin dephasing due to fluctuations of theSi nuclear spin bath, the total spin decoherence rate is expressedassuming a Voigt profile:

$\gamma_{s} = {\frac{\gamma_{s}^{(c)}}{2} + \sqrt{\left( \frac{\gamma_{s}^{(c)}}{2} \right)^{2} + {8\ln 2\left( \frac{1}{T_{2,{nuclear}}^{*}} \right)^{2}}}}$

where T_(2,nuclear)*≈1 μs is the electron-spin dephasing time due tonuclear field fluctuations.

When fitting the data in FIG. 4B, the experimentally determined valueswere used of g_(c)/(2π)=40 MHz and γ_(c)/(2π)=35 MHz, along with thebest-fitting field gradient B_(x) ^(M)=15 mT. For every t_(c), thefitted value for B_(z) is adjusted so that the spin-qubit frequency(E₁−E₀)/h matches the cavity frequency f_(c) exactly. The slightdiscrepancy between theory and experiment for γ_(s) could be due to thefrequency dependence of γ_(c), changes in γ_(c) with B_(z) ^(ext) ext orother decoherence mechanisms that are not captured by this simple model.To resolve such a discrepancy, a complete measurement of γ_(c) as afunction of 2t_(c)/h and the external field B_(z) ^(ext) is needed.

The complete theory also allows g_(s)/(2π) to be calculated for non-zerovalues of ε. Using 2t_(c)/h=7.4 GHz, it was estimated that g_(s)/h=2.3MHz at ε=20 μeV (about 4.8 GHz), close to the value of g_(s)/h=1.0 MHzmeasured at this DQD detuning (FIG. 4A).

In this theoretical model, Purcell decay of the spin qubit through thecavity has been ignored. This is justified because γ_(s) at every valueof t_(c) is measured with a large spin-cavity detuning Δ≈10 g_(s). Theexpected Purcell decay rate of the spin qubit is Γ_(p)/(2π)=[kg_(s)²/(k²/4+Δ²)]/(2π)≈0.02 MHz, well below the measured values ofγ_(s)/(2π). It is also noted that, at least in the 2t_(c)»E_(Z) limit,spin decoherence at ε=0 is dominated by noise-induced dephasing ratherthan by energy relaxation. This is because at 2t_(c)/h=11.1 GHz the spindecoherence rate γ_(s)/(2π)=0.41 MHz corresponds to a coherence time ofT₂=0.4 μs«2T₁=6.4 μs.

Line shapes of vacuum Rabi splittings. In contrast to charge-photonsystems, the two resonance modes in the vacuum Rabi splittings (e.g.,FIGS. 3B-C) show slightly unequal widths. This effect can be seen bycomparing the observed spectrum of DQD1 with the expected behaviour ofan equivalent two-level charge qubit that is coupled strongly to acavity, calculated using a master-equation simulation with thermalphoton number n_(th)=0.02 (black dashed line in FIG. 12A). The unequalwidths are unlikely to be a result of a large thermal photon number inthe cavity, because the transmission spectrum calculated withn_(th)=0.05 (orange dashed line) clearly does not fit the experimentaldata.

Instead, the observed asymmetry probably arises from the dispersiveinteraction between the cavity and the primarily charge-like transitionbetween |0

and |2

, which results in three-level dynamics that is more complicated thanthe two-level dynamics that characterizes charge-photon systems. Here iscompared the experimental observation with theory by calculating A(f)/A₀using g_(c)/(2π)=40 MHz (DQD1) or g_(c)/(2π)=37 MHz (DQD2),γ_(c)/(2π)=105 MHz (DQD1) or γ_(c)/(2π)=130 MHz (DQD2),k/(2π)=1.8 MHz,tunnel couplings 2t_(c)/h=7.4 GHz, B_(x) ^(M)=15 mT and B_(z)=209.6 mT.The results are shown as black solid lines alongside experimental datain FIG. 12A-Bs. The agreement between experiment and theory is very goodfor both devices. In particular, the asymmetry between the vacuum Rabimodes is also seen in the theoretical calculations. The larger values ofγ_(c) used in the theoretical calculations may again be due to thefrequency dependence of γ_(c) or to changes in γ_(c) with B_(z) ^(ext).

Long-range spin-spin coupling. The coherent spin-photon interface may bereadily applied to enable spin-spin coupling across the cavity bus. Hereare evaluated two possible schemes for implementing such a coupling,both of which have been demonstrated with superconducting qubits. Thefirst approach uses direct photon exchange to perform quantum-statetransfer between two qubits. The transfer protocol starts by tuningqubit 1 into resonance with the unpopulated cavity for a time1/(4g_(s)), at the end of which the state of qubit 1 is transferredcompletely to the cavity. Qubit 1 is then detuned rapidly from thecavity and qubit 2 is brought into resonance with the cavity for a time1/(4g_(s)), at the end of which the state of qubit 1 is transferredcompletely to qubit 2. Therefore, the time required for quantum-statetransfer across the cavity is 1/(2g_(s)). Because the decay of vacuumRabi oscillations occurs at a rate k/2+γ_(s), the threshold forcoherent-state transfer between two spin qubits is 2g_(s)/(k/2+γ_(s))>1.The ratio 2g_(s)/(k/2+γ_(s)) is plotted as a function of 2t_(c)/h inExtended Data FIG. 8 a . It can be seen that 2g_(s)/(k/2+γ_(s))>1 forall values of 2t_(c)/h, indicating that spin-spin coupling via realphoton exchange is achievable and may be implemented at any value oft_(c). For our device parameters, the regime 2t_(c)/h≈6 GHz, in whichspin-charge hybridization is large and the ratio 2g_(s)/(k/2+γ_(s))reaches a maximum of 3.5, seems most advantageous for such a couplingscheme.

The second approach to spin-spin coupling uses virtual photon exchange.In this scheme, both spin qubits would operate in the dispersive regime,with an effective coupling rate of J=g_(s) ²(1/Δ₁+1/Δ₂/2, where Δ₁ andΔ₂ are the qubit-cavity detunings for qubits 1 and 2, respectively.Assuming that both qubits operate with an equal detuning Δ_(1,2)=10_(gs)to minimize Purcell decay, J=g_(s)/10. For coherent spin-spininteraction, J>γ_(s) needs to be satisfied, leading to the conditiong_(s)/γ_(s)>10. FIG. 13B is a plot of the ratio g_(s)/γ_(s) as afunction of 2t_(c)/h, observing a maximum of g_(x)/γ_(s)≈4 at2t_(c)/h≈10 GHz. Because the dominant spin mechanism is probablyhyperfine-induced dephasing by the Si nuclei in this regime (thedecoherence rate γ_(s)/(2π)≈0.4 is close to the decoherence ratescommonly found with single-spin qubits in natural Si), transitioning toisotopically purified Si host materials is likely to lead to anorder-of-magnitude reduction in γ_(s)/(2π), as demonstrated recently.Such an improvement will allow virtual-photon-mediated spin-spincoupling to be implemented in our device architecture as well.

Both coupling approaches will benefit substantially from larger valuesof the charge-photon coupling rate g_(c), which is achievable throughthe development of higher-impedance cavities. The superconducting cavityused here is estimated to have an impedance between 200Ω and 300Ω.Increasing this value to about 2 kΩ, which is possible by using NbTiN asthe superconducting material, will lead to another factor-of-threeincrease in g_(c) and therefore g_(s). This could enable theg_(s)/γ_(s)>100 regime to be accessed, where high-fidelity two-qubitgates can be implemented between distant spins. Improvements in thefidelity of cavity-mediated two-qubit gates, particularly in the case ofreal photon exchange, can also be sought by improving the quality factorof the cavity (and thereby reducing k). This is achievable byimplementing stronger gate line filters and removing lossy dielectricssuch as the atomic-layer-deposited Al₂O₃ underneath the cavity.

The following provides additional information regarding the figuresdescribed above. FIGS. 2A-E show an example spin-photon interface. FIG.2A shows optical image of the superconducting microwave cavity. Theinset shows an optical image of the centre pin (0.6 μm) and vacuum gap(20 μm) of the cavity. FIG. 2B shows a false-colour scanning electronmicrograph (SEM) of a DQD. Gate electrodes are labelled as G1, G2, S, D,B1, P1, B2, P2 and B3, where G1 and G2 are screening gates, S and D areused for accumulating electrons in the source and drain reservoirs, andB1 and B3 control the tunnel barrier of each dot to its adjacentreservoir. The locations of the cobalt micromagnets are indicated by theorange dashed lines. FIG. 2C shows a schematic cross-sectional view ofthe DQD device. The blue line indicates the electrostatic confinementpotential which delocalizes a single electron between the two dots(indicated as half-filled circles). The quantization axis of theelectron spin (red arrow) changes between the dots. FIG. 2D shows cavitytransmission amplitude A/A₀ at f=f_(c), where f_(c) is the centrefrequency of the cavity, near the (1, 0)↔(0, 1) inter-dot transition forDQD1, plotted as a function of the voltages on gates P1 and P2, V_(P1)and V_(P2), with B_(z) ^(ext)=0 and V_(B2)=710 mV. The dashed arrowdenotes the DQD detuning parameter ε, which is equal to the differencein the chemical potentials of the two dots and points along the verticaldirection because in this work V_(P1) is changed to vary ε. V_(B2)denotes the voltage on gate B2, which controls the inter-dot tunnelcoupling tc. FIG. 2E shows A/A₀ as a function of ε with V_(B2)=710 mV(red line) and a fit to cavity input-output theory (black dashed line),with g_(c)/(2π)=40 MHz. FIG. 2F shows 2t_(c)/h as a function of V_(B2)for DQD1, obtained by measuring A(ε)/Δ₀ at different values of V_(B2).

FIGS. 3A-C show strong single spin-photon coupling. FIG. 3A shows A/A₀as a function of the cavity drive frequency f and an externally appliedmagnetic field B_(z) ^(ext) for DQD1. Insets show data from DQD2 at thesame values of tc and E and plotted over the same range of f. B_(z)^(ext) ranges from −94 mT to −91.1 mT (91.1 mT to 94 mT) for the left(right) inset. FIG. 3B shows A/A₀ as a function of ƒ for DQD1 at B_(z)^(ext)=90.3 mT (red) and B_(z) ^(ext)=92.2 mT (blue) FIG. 3C shows A/A₀as a function of ƒ for DQD2 at B_(z) ^(ext)=91.1 mT (red) and B_(z)^(ext)=92.6 mT (blue). In FIG. 3B and FIG. 3C, the frequency differencebetween the two transmission peaks, indicated by the black arrows, is11.0 MHz (FIG. 3B) and 10.6 MHz (FIG. 3C). The spin-photon coupling rategs/(2π) corresponds to half the frequency separation and so is 5.5 MHzfor DQD1 and 5.3 MHz for DQD2.

FIGS. 4A-B show Electrical control of spin-photon coupling. FIG. 4Ashows A/A₀ as a function of f and B_(z) ^(ext) at ε=0 (left), ε=20 μeV(about 4.8 GHz; middle) and ε=40 μeV (about 9.7 GHz; right), with2t_(c)/h=7.4 GHz. Insets show A/A₀ as a function of ƒ at the values ofB_(z) ^(ext) indicated by the white dashed lines in the main panels.FIG. 4B shows spin-photon coupling rate gs/(2π) (top) and spindecoherence rate γ_(s)/(2π) (bottom) as functions of 2t_(c)/h, with ε=0(data). The dashed lines show theoretical predictions. A potentialuncertainty of 0.01-0.1 MHz exists for each value of g_(s)/(2π) andγ_(s)/(2π) owing to uncertainties in the locations of the transmissionpeaks used to determine g_(s)/(2π) (e.g., FIG. 10A-B) and the widths ofthe Lorentzian fits used to determine γ_(s)/(2π) (e.g., FIGS. 9A-D).FIG. 4C shows DQD energy levels as a function of ε, calculated withB_(z) ^(ext)+B_(z) ^(M)=209 mT, B_(x) ^(M)=15 mT and 2t_(c)/h=7.4 GHz.Here B_(z) ^(M) denotes the magnetic field produced by the cobaltmicromagnet parallel to B_(z) ^(ext), and B_(x) ^(M) is related to thestrength of the inhomogeneous magnetic field perpendicular to B_(z)^(ext). The symbols ↑ (↓), L (R) and −(+) denote the quantum states ofthe electron that correspond to up (down) spin states, left-dot(right-dot) orbital states and molecular bonding (anti-bonding) states,respectively. The schematics at the top illustrate the distribution ofthe wavefunction of the electron in different regimes of ε. For ε»t_(c)and −ε»t_(c), the electron is localized within one dot and tunnellingbetween the dots is largely forbidden, resulting in a small gs due to asmall effective oscillating magnetic field. For |ε|«tc, the electron maytunnel between the two dots and experience a large oscillating magneticfield due to the spatial field gradient, resulting in a large g_(s).

FIGS. 5A-D shows quantum control and dispersive readout of a singlespin. FIG. 5A shows cavity phase response Δϕ at f=f_(c) when gate P1 isdriven continuously at a variable frequency f_(s) and power P_(s)=−106dBm, with 2t_(c)/h=9.5 GHz and ε=0. A background phase response,obtained by measuring Δϕ(B_(z) ^(ext)) in the absence of a microwavedrive on P1, is subtracted from each column of the data to correct forslow drifts in the microwave phase. FIG. 5B shows electron spinresonance (ESR) line as measured in Δϕ(f_(s)) at 2t_(c)/h=11.1 GHz, ε=0,B_(z) ^(ext)=92.18 mT and P_(s)=−123 dBm (data). The dashed line shows afit to a Lorentzian with a full-width at half-maximum ofγ_(s)/π=0.81±0.04 MHz (indicated by the arrows). FIG. 5C shows schematicshowing the experimental sequence for coherent spin control andmeasurement. Spin control is performed using a high-power microwaveburst when the electron is largely localized within one dot (|ε|»t_(c))and spin-photon coupling is turned off. Spinstate readout is achievedusing the dispersive response of a cavity photon at ϵ=0 and whenspin-photon coupling is turned on. FIG. 5D shows Δϕ as a function ofτ_(B), with 2t_(c)/h=11.1 GHz and ε′=70 μeV, showing single-spin Rabioscillations. The excited-state population of the spin qubit P_(↑) isindicated on the right y axis (see Methods).

FIG. 6 shows micromagnet design. To-scale drawing of the micromagnetdesign, superimposed on top of the SEM image of the DQD. The coordinateaxes and the direction of the externally applied magnetic field B_(z)^(ext) are indicated at the bottom. In this geometry, the DQD electronexperiences a homogeneous z field B_(z)≈B_(z) ^(ext)+B_(z) ^(M). Thetotal x field B_(x) that is experienced by the electron is spatiallydependent, being approximately B_(x,L) ^(M) (B_(x,R) ^(M)) when theelectron is in the L (R) dot (ε»t_(c)) and (B_(x,L) ^(M)+B_(x,R) ^(M))/2when the electron is delocalized between the DQDs (ε=0). The y fieldB_(y) for the DQD electron is expected to be small compared to the otherfield components for this magnet design.

FIG. 7 shows photon number calibration. The ESR resonance frequencyf_(ESR), measured using the phase response of the cavity Δϕ in thedispersive regime (e.g., FIG. 5B), is plotted as a function of theestimated power at the input port of the cavity P (data). The device isconfigured with g_(s)/(2π)=2.4 MHz and spin-photon detuning Δ/2π≈−18MHz≈The dashed line shows a fit to f_(ESR)=f_(ESR)(P=0)+(2n_(ph)g_(s)²/(Δ)/(2π), where n_(ph) is the average number of photons in the cavity,plotted as the top x axis. The experiments are conducted with P≈−133 dBm(0.05 fW), which corresponds to n_(ph)≈0.6. The error bars indicate theuncertainties in the centre frequency of the ESR transition.

FIGS. 8A-D show DQD stability diagrams. The cavity transmissionamplitude A/A₀ (FIGS. 8A, 8C) and phase response Δϕ (FIG. 8B, 8D) areplotted as functions of V_(P1) and V_(P2) for DQD1 (FIGS. 8A, 8B) andDQD2 (FIGS. 8C, 8D), obtained with f=f_(c). The (1, 0)↔(0, 1)transitions are clearly identified on the basis of these measurementsand subsequently tuned close to resonance with the cavity for theexperiments described in the main text. The red circles indicate thelocations of the (1, 0)↔(0, 1) transitions of the two DQDs.

FIGS. 9A-D show spin decoherence rates at different DQD tunnelcouplings. ESR line, as measured in the cavity phase response Δϕ(f_(s)),is shown for different values of 2t_(c)/h in the low-power limit (data).ε=0 for every dataset. Dashed lines are fits with Lorentzian functionsand γ_(s)/(2π) is determined as the half-width at half-maximum of eachLorentzian. The spin-photon detuning |Δzz|≈g_(s) for each dataset, toensure that the system is in the dispersive regime.

FIGS. 10A-B show spin-photon coupling strengths at different DQD tunnelcouplings. FIG. 10A shows vacuum Rabi splittings for 2t_(c)/h<f_(c) andFIG. 10B shows vacuum Rabi splittings for 2t_(c)/h>f_(c), obtained byvarying B_(z) ^(ext) until a pair of resonance peaks with approximatelyequal heights emerges in the cavity transmission spectrum A/A₀.g_(s)/(2π) is then estimated as half the frequency difference betweenthe two peaks. ε=0 for every dataset. g_(s) is difficult to measure for5.2 GHz<2t_(c)/h<6.7 GHz owing to the small values of A/A₀ that arisefrom the large spin decoherence rates γ_(s) in this regime.

FIG. 11 shows spin relaxation at ε=0. The time-averaged phase responseof the cavity Δϕ is shown as a function of wait time T_(M) (data),measured using the pulse sequence illustrated in FIG. 5C. The microwaveburst time is fixed at τ_(B)=80 ns. The dashed line shows a fit usingthe function ϕ₀+ϕ₁(T₁/T_(M))[1−exp(−T_(M)/T₁)], which yields a spinrelaxation time of T₁≈3.2 μs. The experimental conditions are the sameas for FIG. 5D.

FIGS. 12A-B show theoretical fits to vacuum Rabi splittings. Thecalculated cavity transmission spectra (black solid lines) aresuperimposed on the experimentally measured vacuum Rabi splittings shownin FIG. 3B-C (data). The calculations are produced with g_(c)/(2π)=40MHz (g_(c)/(2π)=37 MHz), κ/(2π)=1.8 MHz, γ_(c)/(2π)=105 MHz(γ_(c)/(2π)=120 MHz), B_(z)=B_(z) ^(ext)+B_(z) ^(M)=209 mT, B_(x)^(M)=(B_(x,R) ^(M)−B_(x,L) ^(M))/2=15 mT and 2t_(c)/h=7.4 GHz for DQD1(DQD2). For comparison, A(f)/A₀, simulated for a two-level charge qubitwith a decoherence rate of γ_(c)/(2π)=2.4 MHz coupled to a cavity withκ/(2π)=1.8 MHz at a rate g_(c)/(2π)=5.5 MHz, is shown in a for thermalphoton numbers of n_(th)=0.02 (black dashed line) and n_(th)=0.5 (reddashed line).

FIGS. 13A-B shows prospects for long-range spin-spin coupling. FIG. 13Ashows the ratio 2g_(s)/(κ/2+γ_(s)) as a function of 2t_(c)/h, calculatedusing the data in FIG. 4B and κ/(2π)=1.8 MHz. FIG. 13B shows the ratiog_(s)/γ_(s) as a function of 2t_(c)/h, also calculated using the data inFIG. 4B.

Second Section: Input-Output Theory for Spin-Photon Coupling in SiDouble Quantum Dots

The interaction of qubits via microwave frequency photons enableslong-distance qubit-qubit coupling and facilitates the realization of alarge-scale quantum processor. However, qubits based on electron spinsin semiconductor quantum dots have proven challenging to couple tomicrowave photons. In this section is shown that a sizable coupling fora single electron spin is possible via spin-charge hybridization using amagnetic field gradient in a silicon double quantum dot. Based onparameters already shown in recent experiments, optimal working pointsare predicted to achieve a coherent spin-photon coupling, an essentialingredient for the generation of long-range entanglement. Furthermore,input-output theory is employed to identify observable signatures ofspin-photon coupling in the cavity output field, which may provideguidance to the experimental search for strong coupling in suchspin-photon systems and opens the way to cavity-based readout of thespin qubit.

Building a practical solid state quantum processor may rely on aflexible scheme of coupling individual qubits such that a 2D array ofqubits, or even a network with connectivity between arbitrary pairs ofqubits (“all-to-all” connectivity), may be achieved. For superconductingqubits, entanglement of qubits separated by macroscopic distances hasbeen demonstrated using the approach of circuit quantum electrodynamics(cQED), whereby photons confined inside microwave frequency cavitiesserve as mobile carriers of quantum information that mediate long-rangequbit interactions. Compared to superconducting qubits, qubits based onspins of electrons in semiconductor quantum dots (QDs) have the virtueof long lifetimes (T₁) that can be on the order of seconds for Si. Onthe other hand, the coupling between electron spins is typically basedon nearest neighbor exchange interactions; therefore the coupling ofspin qubits has remained limited to typical distances <100 nm. Thedevelopment of a spin-cQED architecture in which spin qubits arecoherently coupled to microwave frequency photons is therefore animportant goal which would enable a spin-based quantum processor withfull connectivity.

To transfer quantum states between a spin qubit and a cavity photon withhigh fidelity, it is important to achieve the strong-coupling regime inwhich the spin-photon coupling g_(s) exceeds both the cavity decay rateκ and the spin decoherence rate γ_(s). While demonstrations of strongcoupling have already been made with superconducting qubits andsemiconductor charge qubits, such a task has proven challenging for asingle spin due to its small magnetic dipole, which results in couplingrates that are less than 1 kHz and too slow compared to typical spindephasing rates. An alternative route toward strong spin-photon couplinginvolves hybridizing the spin and charge states of QD electrons. Therelatively large electric susceptibilities of the electron charge stateslead to an effective spin-photon coupling rate g_(s) on the order ofMHz, as recently demonstrated by a carbon nanotube double quantum dot(DQD) device. However, spin-charge hybridization also renders spinqubits susceptible to charge noise, which has up to now prevented thestrong coupling regime from being reached with a single spin. Onlyrecently, the achievement of strong coupling between single spins andmicrowave photons has been reported. Coupling of three-electron spinstates to photons has also been reported.

Here is provided an analysis of a scheme for strong spin-photon couplingusing a semiconductor DQD placed in the inhomogeneous magnetic field ofa micromagnet. This disclosure further predicts a complete map of theeffective spin-photon coupling rate g_(s) and spin decoherence rateγ_(s). This allows for finding optimal working points for coherentspin-photon coupling. Further presented are detailed calculations of thecavity transmission and identify experimentally observable signatures ofspin-photon coupling. It is predicted that the strong-coupling regimebetween a single spin and a single photon is achievable in Si usingvalues of the charge-cavity vacuum Rabi frequency g_(c) and chargedecoherence rate γ_(c) from recent experiments.

The physical system consists of a gate-defined Si DQD that is embeddedin a superconducting cavity; see FIG. 14 . The electric-dipoleinteraction couples the electronic charge states in the DQD to thecavity electric field. The introduction of an inhomogeneous magneticfield, as sketched in FIG. 14 , hybridizes the charge states of a DQDelectron with its spin states, indirectly coupling the cavity electricfield to the electron spin.

Model. It is assumed that the DQD is filled with a single electron andhas two charge configurations, with the electron located either on theleft (L) or right (R) dot, with on-site energy difference (detuning) ϵand tunnel coupling t_(c). If a homogeneous magnetic field B_(z) and aperpendicular spatial gradient field B_(x) are applied one can model thesingle electron DQD with the HamiltonianH ₀=½(ϵτ_(z)+2t _(c)τ_(x) +B _(z)σ_(z) +B _(x)σ_(x)τ_(z)),   (1)where τ_(α) and σ_(α) are the Pauli operators in position (L,R) and spinspace, respectively. Here, B_(z(x)) are the magnetic fields in energyunits and ℏ=1. The valley degree of freedom present in this type of DQDshas not been considered in this model. Low-lying valley states areexpected to lead to additional resonances and be detected in the cavitytransmission. For the purposes concerning this work, spin-photoncoupling via spin-charge hybridization, the ideal situation is to havequantum dots with a sufficiently large valley splitting, ≳40 μeV. Thereis therefore a four-level Hamiltonian with eigenenergies E_(n) andeigenstates |n

for n=0, . . . , 3. The eigenenergies in the regime2t_(c)>B_(z)(2t_(c)<B_(z)) are shown in FIG. 15A. As shown in FIG. 15B,the magnetic field gradient generates spin-charge hybridization,coupling the original (B_(x)=0) energy levels (dashed lines) andinducing anticrossings at ϵ=±√{square root over (B_(z) ²−4t_(c) ²)} if2t_(c)<B_(z).

In the dipole approximation, the coupling of the DQD to the electricfield of a microwave cavity can be described asH ₁ =g _(c)(a+a ^(†))τ_(z),  (2)where a and a^(†) are the bosonic cavity photon creation andannihilation operators. The Hamiltonian for the relevant cavity mode,with frequency ω_(c), is H_(c)=ω_(c)a_(†)a. In the eigenbasis of H₀, theinteraction acquires nondiagonal elements,

$\begin{matrix}{H_{I} = {{g_{c}\left( {a + a^{\dagger}} \right)}{\sum\limits_{n,{m = 0}}^{3}{d_{nm}\left. ❘n \right\rangle{\left\langle m❘ \right..}}}}} & (3)\end{matrix}$

As shown below, the essential dynamics of this system can also bedescribed in terms of a so-called Λ system, with two weakly coupledexcited states and a ground state; see FIGS. 15C-D.

FIG. 14 shows a schematic illustration of the Si gate-defined DQDinfluenced by an homogeneous external magnetic field, B_(z), and theinhomogeneous perpendicular magnetic field created by a micromagnet,with opposite direction at the positions of the two QDs, ±B_(x). The DQDis electric-dipole-coupled to the microwave cavity represented in blue.The cavity field is excited at the left and right ports via a_(in,1) anda_(in,2), and the output can be measured either at the left (a_(out,1))or right port (a_(out,2)).

Input-output Theory. To treat the DQD and the cavity as an open system,the Heisenberg picture is considered using the quantum Langevinequations (QLEs) for the system operators, including the photonoperators a,a† and σnm=|n

m|. This treatment enables the calculation of the outgoing fields,a_(out,1) and a_(out,2), at the two cavity ports given the incoming weakfields, ain,1 and a_(in,2).

FIGS. 15A-B show energy levels En(n=0, . . . , 3) as a function of theDQD detuning parameter ϵ. The dashed lines are the energy levels withouta magnetic field gradient (B_(x)=0). They correspond to the bonding (−)and antibonding (+) orbitals with spin in the z direction, denoted by|±, ↑(↓)). The arrow represents the transition driven by the probefield, at frequency ωR. Here, the parameters B_(z)=24 μeV and Bx=10 μeVare chosen. For the tunnel coupling: FIG. 15A shows t_(c)=15.4μeV>B_(z)/2 and FIG. 15B t_(c)=10.2 μeV>B_(z)/2.

FIGS. 15C-D show schematic representation of the ∧ system that capturesthe essential dynamics in FIG. 15A-B, respectively (near ϵ=0). If theorbital energy, Ω=√{square root over (ϵ²+4t_(c) ²)}, is near B_(z), thelevels |−, ↑

and |+, ↓

hybridize into the states |1

and |2

due to the magnetic field gradient, while the ground state isapproximately unperturbed |0

˜|−, ↓

. The wavy lines represent charge decoherence with rate γ_(c).

If the average population of the energy levels, p_(n)≡

σ_(nn)

, follows a thermal distribution, the linear response to a probe fieldis reflected in the dynamics of the nondiagonal operators σ_(nm). If thecavity is driven with a microwave field with a near-resonant frequencyω_(R), the QLEs in a frame rotating with the driving frequency read

$\begin{matrix}{{\overset{.}{a} = {\overset{.}{{\iota\Delta}_{0}a} - {\frac{\kappa}{a}a} + {\sqrt{\kappa_{1}}a_{{in},1}} + {\sqrt{\kappa_{2}}a_{{in},2}} - {{ig}_{c}e^{\overset{.}{i}\omega_{R}t}{\sum\limits_{n,{m = 0}}^{3}{d_{nm}\sigma_{nm}}}}}},} & (4)\end{matrix}$ $\begin{matrix}{{{\overset{.}{\sigma}}_{nm} = {{{- {i\left( {E_{m} - E_{n}} \right)}}\sigma_{nm}} - {\sum\limits_{n^{\prime}m^{\prime}}{\gamma_{{nm},{n^{\prime}m^{\prime}}}\sigma_{n^{\prime}m^{\prime}}}} + {\sqrt{2\gamma}F} - {{{ig}_{c}\left( {{ae}^{{- i}\omega_{R}t} + {a^{\dagger}e^{i\omega_{R}t}}} \right)}{d_{mn}\left( {p_{n} - p_{m}} \right)}}}},} & (5)\end{matrix}$where Δ₀=ω_(R)−ω_(c) is the detuning of the driving field relative tothe cavity frequency and κ is the total cavity decay rate, with κ_(1,2)the decay rates through the input and output ports.

is the quantum noise of the DQD and a_(in,i) denote the incoming partsof the external field at the ports. The outgoing fields can becalculated as a_(out,i)=√{square root over (κ_(i))}a−a_(in,i). Thesuperoperator γ, with matrix elements γ_(nm,n′m′), represents thedecoherence processes which, in general, can couple the equations forthe operators σ_(nm). In this work, the decoherence superoperator γ willcapture charge relaxation and dephasing due to charge noise (e.g., see“DECOHERENCE MODEL” section below), since these are the most relevantsources of decoherence.

This formalism allows computation of the transmission through themicrowave cavity. Within a rotating-wave approximation (RWA) (e.g., see“MULTILEVEL RWA” section below) the explicit time dependence in Eqs. (4)and (5) can be eliminated to solve the equations for the expected valueof these operators in the stationary limit (ā,σ _(n,m)) to obtain thesusceptibilities,σ _(n,n+j)=χ_(n,n+j) ā (j=1, . . . ,3−n),   (6)and the transmission A=ā_(out,2)/ā_(in,1),

$\begin{matrix}{{A = \frac{{- i}\sqrt{\kappa_{1}\kappa_{2}}}{{- \Delta_{0}} - {i{\kappa/2}} + {g_{c}{\sum\limits_{n = 0}^{2}{\sum\limits_{j = 1}^{3 - n}{d_{n,{n + j}}\chi_{n,{n + j}}}}}}}},} & (7)\end{matrix}$which is in general a complex quantity. Considered here are

a_(in,2)

=0 and

=0

Orbital Basis. In the product basis of antibonding and bonding orbitals+, − with spin ↑↓ in the z direction, {|+, ↑

, |−, ↑

, |+, ↓

, |−, ↓

}, the Hamiltonian in Eq. (1) reads

$\begin{matrix}{{H_{0}^{orb} = {\frac{1}{2}\begin{pmatrix}{\Omega + B_{z}} & 0 & {B_{x}\sin\theta} & {{- B_{x}}\cos\theta} \\0 & {{- \Omega} + B_{z}} & {{- B_{x}}\cos\theta} & {{- B_{x}}\sin\theta} \\{B_{x}\sin\theta} & {{- B_{x}}\cos\theta} & {\Omega - B_{z}} & 0 \\{{- B_{x}}\cos\theta} & {{- B_{x}}\sin\theta} & 0 & {{- \Omega} - B_{z}}\end{pmatrix}}},} & (8)\end{matrix}$where Ω=√{square root over (ϵ²+4t_(c) ²)} is the orbital energy and

$\theta = {\arctan\frac{\epsilon}{2t_{c}}}$is introduced as the “orbital angle.” In this basis the dipole operatortakes the form

$\begin{matrix}{d^{orb} = {\begin{pmatrix}{\sin\theta} & {{- \cos}\theta} & 0 & 0 \\{{- \cos}\theta} & {{- \sin}\theta} & 0 & 0 \\0 & 0 & {\sin\theta} & {{- \cos}\theta} \\0 & 0 & {{- \cos}\theta} & {{- \sin}\theta}\end{pmatrix}.}} & (9)\end{matrix}$In the simplest case, ϵ=0, the orbital angle θ is zero, and one canrewrite the Hamiltonian as

$\begin{matrix}{{{H_{0}^{orb}\left( {\epsilon = 0} \right)} = {\frac{r}{2}\begin{pmatrix}\frac{{2t_{c}} + B_{z}}{r} & 0 & 0 & {{- \sin}\Phi} \\0 & {{- \cos}\Phi} & {{- \sin}\Phi} & 0 \\0 & {{- \sin}\Phi} & {\cos\Phi} & 0 \\{{- \sin}\Phi} & 0 & 0 & \frac{{{- 2}t_{c}} - B_{z}}{r}\end{pmatrix}}},} & (10)\end{matrix}$with r=√{square root over ((2t_(c)−B_(z))²+B_(x) ²)} and the spin-orbitmixing angle

$\Phi = {\arctan{{\frac{B_{x}}{{2t_{c}} - B_{z}}\left\lbrack {\Phi \in \left( {0,\pi} \right)} \right\rbrack}.}}$As the dipole operator couples the states |−, ↓

and |+, ↓

and the field gradient couples |+, ↓

to |−, ↑

, the combination of these two effects leads to a coupling between thetwo different spin states |−, ↓

and |−, ↑

. It is this coupling that can be harnessed to coherently hybridize asingle electron spin with a single photon and achieve thestrong-coupling regime.

Results.

A. Effective coupling at zero detuning. The spin-charge hybridizationcreated by the inhomogeneous magnetic field allows for the coupling ofthe spin to the cavity. This is visible in the form of the operator d inthe eigenbasis; see Eq. (3). In the simple case of zero DQD detuning,ϵ=0, the ordered energy levels areE _(3,0)=±½√{square root over ((2t _(c) +B _(z))² +B _(x) ²,)}   (11)E _(2,1)=±½√{square root over ((2t _(c) +B _(z))² +B _(x) ²)}.   (12)

Using the spin-orbit mixing angle Φ the eigenstates |1

and |2

can be expressed as

$\begin{matrix}{{\left. ❘1 \right\rangle = {{\cos\frac{\Phi}{2}\left. ❘{- \left. ,\uparrow \right.} \right\rangle} + {\sin\frac{\Phi}{2}\left. ❘{+ \left. ,\downarrow \right.} \right\rangle}}},} & (13)\end{matrix}$ $\begin{matrix}{{\left. ❘2 \right\rangle = {{\cos\frac{\Phi}{2}\left. ❘{- \left. ,\uparrow \right.} \right\rangle} - {\cos\frac{\Phi}{2}\left. ❘{+ \left. ,\downarrow \right.} \right\rangle}}},} & (14)\end{matrix}$while the other two can be approximated by|0

≃|−,↓

  (15)|3

≃|+,↑

,  (16)if r«(2t_(c)+B_(z)), i.e., for small |2t_(c)−B_(z)|. In this limit, thedipole matrix elements,

$\begin{matrix}{{d = \begin{pmatrix}0 & d_{01} & d_{02} & 0 \\d_{01} & 0 & 0 & d_{13} \\d_{02} & 0 & 0 & d_{23} \\0 & d_{13} & d_{23} & 0\end{pmatrix}},} & (17)\end{matrix}$simplify to

$\begin{matrix}{{d_{01} = {d_{23} \simeq {{- \sin}\frac{\Phi}{2}}}},} & (18)\end{matrix}$ $\begin{matrix}{{d_{02} = {{- d_{13}} \simeq {\cos\frac{\Phi}{2}}}},} & (19)\end{matrix}$

This means that the hybridization due to the weak magnetic fieldgradient generates an effective coupling between the levels |−, ↓

and |−, ↑

, with opposite spin. The spin nature of the transitions 0↔1 and 0↔2depends on the spin-orbit mixing angle Φ; see Eqs. (13) to (15).Assuming that the cavity frequency is tuned to the predominantlyspinlike transition, which is 0↔1 (0↔2) for cos Φ>0 (cos<0), asindicated in the level structure of FIGS. 15C-D, the effectivespin-cavity coupling strength will be given by g_(s)=g_(c)|d₀₁₍₂₎|.

B. Effective coupling at ϵ≠0. For ϵϵ0 the energy levels areE _(3,0)=±½[(Ω+√{square root over (B _(z) ² +B _(x) ² sin²θ)})² +B _(x)² cos²θ]^(1/2),  (20)E _(2,1)=±½[(Ω−√{square root over (B _(z) ² +B _(x) ² sin²θ)})² +B _(x)² cos²θ]^(1/2).  (21)Analogous to the previous section, if √{square root over((Ω−B_(z))²+B_(z) ²)}«(Ω+B_(z)) one can approximate the eigenstates byEqs. (13) to (16) where the spin-orbit mixing angle is now

$\Phi = {\arctan{{\frac{B_{x}\cos\theta}{\Omega - B_{z}}\left\lbrack {\Phi \in \left( {0,\pi} \right)} \right\rbrack}.}}$Within this approximation,

$\begin{matrix}{{d_{01} = {d_{23} \simeq {{- \cos}\theta\sin\frac{\Phi}{2}}}},} & (22)\end{matrix}$ $\begin{matrix}{d_{02} = {{- d_{13}} \simeq {\cos\theta\cos{\frac{\Phi}{2}.}}}} & (23)\end{matrix}$

C. Effective coupling map. Before calculating the effect of the Si DQDon the cavity transmission A, one can estimate the magnitude of thecoupling g_(s). For Ω>B_(z)(Ω<B_(z)), 0↔1 (0↔2) is predominantly a spintransition; therefore, one can obtain a map for the effective couplingby using g_(s)=g_(c)|d₀₁₍₂₎|; see FIG. 16A. As the value of Ω approachesB_(z), Φ tends to π/2 and the coupling is maximized. However, in thisregime, due to strong spin-charge hybridization, the charge nature ofthe transition increases (e.g., see Eqs. (13) and (14)) and with it thedecoherence rate increases, preventing the system from reaching strongcoupling. In the following it is shown that the ratio of the couplingrate to the total decoherence rate can be optimized by working away frommaximal coupling. In particular the strong-coupling regime for the spincan be achieved.

As described further in the “GENERAL MAGNETIC FIELD GRADIENT DIRECTION”section below, the model is generalized to a less symmetric situation,where the gradient of the magnetic field between the quantum dotpositions is not perpendicular to the homogeneous magnetic field. Thiseffect is not critical and it is expected that the strong-couplingregime to be accessible as well. For simplicity, the symmetric case isconsidered in the following.

D. Cavity transmission. In the following, the DQD is considered to be inits ground state, such that p_(n)=δ_(n,0) in Eq. (5). If the cavityfrequency is close to the Zeeman energy, ω_(c)≈B_(z), the transition 0↔3is off resonant and the relevant dynamics is contained in the levelstructure of FIGS. 15C and 2 (d). Moreover, this transition is notcoupled to the others since d₀₃=0 and γ_(03,nm)˜δ_(n0)δ_(m3) (e.g., see“DECOHERENCE MODEL” section below)). To calculate the cavity response,it is sufficient to solve the QLEs for

a

,

σ₀₁

, and

σ₀₂

(in the following the brackets are omitted) within the RWA (e.g., see“MULTILEVEL RWA” section below).

As explained above, the decoherence processes accounted for in Eq. (5)can result in a different decay rate for every transition and can alsocouple different transitions. As shown in the section entitled“DECOHERENCE MODEL” below, the decoherence superoperator in the basis{σ₀₁, σ₀₂} reads

$\begin{matrix}{{\gamma = {\gamma_{c}\begin{pmatrix}{\sin^{2}\frac{\Phi}{2}} & {- \frac{\sin\Phi}{2}} \\{- \frac{\sin\Phi}{2}} & {\cos^{2}\frac{\Phi}{2}}\end{pmatrix}}},} & (24)\end{matrix}$where γ_(c) is the total charge dephasing rate, which accounts forcharge relaxation and dephasing due to charge noise. With this, the QLEsread

$\begin{matrix}{{\overset{.}{a} = {{i\Delta_{0}a} - {\frac{\kappa}{2}a} + {\sqrt{\kappa_{1}}a_{{in},1}} - {{ig}_{c}\left( {{d_{01}\sigma_{01}} + {d_{02}\sigma_{02}}} \right)}}},} & (25)\end{matrix}$ $\begin{matrix}{{{\overset{.}{\sigma}}_{01} = {{{- i}\delta_{1}\sigma_{01}} - {\gamma_{c}\sin^{2}\frac{\Phi}{2}\sigma_{01}} + {\frac{\gamma_{c}}{2}\sin{\Phi\sigma}_{02}} - {{ig}_{c}{ad}_{10}}}},} & (26)\end{matrix}$ $\begin{matrix}{{{\overset{.}{\sigma}}_{02} = {{{- i}\delta_{2}\sigma_{02}} - {\gamma_{c}\cos^{2}\frac{\Phi}{2}\sigma_{02}} + {\frac{\gamma_{c}}{2}\sin{\Phi\sigma}_{01}} - {{ig}_{c}{ad}_{20}}}},} & (27)\end{matrix}$with the detunings δn≡E_(n)−E₀−E₀−ω_(R) (n=1,2). The solution of theseequations in the stationary limit allows us to compute thesusceptibilities

$\begin{matrix}{{\chi_{01} = {\frac{{\overset{\_}{\sigma}}_{01}}{\overset{\_}{a}} = \frac{g_{c}\cos\theta{\sin\left( {\Phi/2} \right)}}{\delta_{1} - {i\gamma_{eff}^{(2)}}}}},} & (28)\end{matrix}$ $\begin{matrix}{{\chi_{02} = {\frac{{\overset{\_}{\sigma}}_{02}}{\overset{\_}{a}} = \frac{{- g_{c}}\cos\theta{\cos\left( {\Phi/2} \right)}}{\delta_{2} - {i\gamma_{eff}^{(1)}}}}},} & (29)\end{matrix}$where γ_(eff) ^((n))≡γ_(c)[δ₂ sin²(Φ/2)+δ₁ cos² (Φ/2)]/δ_(n) and thetransmission through the cavity

$\begin{matrix}{{A = \frac{{- i}\sqrt{\kappa_{1}\kappa_{2}}}{{- \Delta_{0}} - {i\frac{\kappa}{2}} + {g_{c}\left( {{\chi_{01}d_{01}} + {\chi_{02}d_{02}}} \right)}}},} & (30)\end{matrix}$with d₀₁ and d₀₂ defined in Eqs. (22) and (23). If 0↔1 (0↔2) ispredominantly a spin transition and the corresponding transition energyis in resonance with the cavity frequency, it is expected to have aneffective spin decoherence rate γ_(s)=γ_(eff) ⁽²⁾(γ_(s)=γ_(eff) ^((y))).In FIG. 16B is shown the ratio γ_(s)/γ_(c), together with g_(s)/g_(c),as a function of the tunnel coupling for ϵ=0. Here, the externalmagnetic field is set to the resonant value B_(z) ^(res) such thatE₁₍₂₎−E₀=ω_(c). This is

$\begin{matrix}{{B_{z}^{res} = {\omega_{c}\sqrt{1 - \frac{B_{x}^{2}}{\omega_{c}^{2} - {4t_{c}^{2}}}}}},} & (31)\end{matrix}$

for ϵ=0. In a small region around 2t_(c)≈ω_(c)[√{square root over(ω_(c)(ω_(c)−B_(x)))}<2t_(c)<√{square root over (ω_(c)(ω_(c)+B_(x)))}],indicated with the vertical lines in FIG. 16B, it is not possible toachieve the desired resonance by tuning B_(z). It is observed that inthe wings of the peak g_(s)/g_(c) »γ_(s)/γ_(c), which may lead thespin-cavity system to be in the strong-coupling regime even when thecharge-cavity system is not (g_(c)<γ_(K).). This is visible in theinset, where it is shown that the ratio g_(s)/√{square root over ([γ_(s)²+(κ/2)²]/2)} exceeds one, signifying the strong-coupling regime (e.g.,see “CHARACTERIZATION OF THE SPIN-PHOTON COUPLING” section). For thiscalculation, and in the remainder of the text, it is assumed that agradient of the magnetic field between the two dots of B_(x)≈15 mT, oneorder of magnitude smaller than the external magnetic field, acharge-cavity coupling g_(c)/2π=40 MHz, a cavity decay rate on the orderof MHz, and a much larger charge dephasing rate on the order of 100 MHz.

FIG. 16A shows expected effective coupling gs/gc=|d01(2)|, according toEqs. (22) and (23) as a function of t_(c) and ϵ. The black dashed linecorresponds to Ω=B_(z) and separates the region where gs/gc=|d0,(1)|(above) from the region where gs/gc=|d0,(2)| (below). The mostinteresting region lies in between the two white dashed lines, where theapproximations are accurate [√{square root over ((Ω−B_(z))²+B_(x)²)}«(Ω+B_(z))]. Chosen values include B_(x)=1.62 μeV and B_(z)=24 μeV.FIG. 16B shows spin-photon coupling strength g_(s)/g_(c) and spindecoherence rate γ_(s)/γ_(c) as a function of t_(c) for ε=0, B_(z)=B_(z)^(res), and B_(x)=1.62 μeV. Between the two blue vertical lines, theresonance cannot be achieved by tuning B_(z). Inset: ratiog_(s)/√{square root over ([γ_(s) ²+(κ/2)²]/2)} in the same range. Thecoupling is strong when this quantity is larger than one (dashed line).The values chosen include γ_(c)/2π=100 MHz, g_(c)/2π=40 MHz, andκ/2π=1.77 MHz.

According to the level structure, it is expected to observe a signatureof spin-photon coupling by driving the cavity near resonance (Δ₀≈0) andsweeping the external magnetic field through the cavity frequency. FIG.17 shows cavity transmission spectrum, |A|, as a function of B_(z) and ϵat zero driving frequency detuning Δ₀=0. The other parameters aret_(c)=15.4 μeV, B_(x)=1.62 μeV, γ_(c)/2π=(100+150|sin θ|) MHz,g_(c)/2π=40 MHz, κ/2π=1.77 MHz, and ω_(c)/2π=5.85 GHz 24 μeV. In FIG. 17is shown the calculated transmission through the cavity as a function ofthe external magnetic field B_(z) and the DQD detuning ϵ when thedriving frequency matches the cavity frequency. Chosen values includeκ₁=κ₂=κ/2. When the cavity frequency is close to the transition energy0↔1, the interaction between the electron and the cavity field resultsin a significantly reduced cavity transmission. Interestingly, close toB_(z)≈ω_(c) the transmission approaches one due to an interferencebetween the two energy levels. At this point, χ₀₁d₀₁+χ₀₂d₀₂≃0.

FIG. 18A shows cavity transmission spectrum |A| as a function of B_(z)and Δ₀. FIG. 18 B-C shows |A| as a function of B_(z) (Δ₀) for the valueof Δ₀ (B_(z)) indicated by the black (blue) dashed line. In (a)t_(c)=15.4 μeV. In FIGS. 18B-C are shown the result for this value(solid line) and for t_(c)=13.9 μeV (dashed line) and t_(c)=19.7 μeV(dotted line). The other parameters are ε=0, B_(x)=1.62 μeV,γ_(e)/2π=100 MHz, g_(c)/27π=40 MHz, κ/2π=1.77 MHz, and ω_(c)/2π=5.85GHz≈24 μeV.

In the usual scenario of a two-level system coupled to a photoniccavity, strong coupling results in light-matter hybridization, asevidenced in the observation of vacuum Rabi splitting in the cavitytransmission spectrum when the qubit transition frequency matches thecavity frequency. The two vacuum Rabi normal modes are separated by afrequency corresponding to the characteristic rate of the light-matterinteraction, and the linewidth of each mode reflects the averagedecoherence rate of light and matter. In FIG. 18A, the absolute value ofthe transmission, |A|, is shown as a function of the magnetic fieldB_(z) and the driving frequency relative to ω_(c), Δ₀, at ϵ=0. The phasegives similar information (not shown). When the driving frequency isnear the cavity frequency, Δ₀≈0, two peaks emerge in the cavitytransmission, signifying the strong-coupling regime. In FIGS. 18B-C areshown the horizontal and vertical cuts of this figure at Δ₀=0 andB_(z)=B_(z) ^(res), respectively, where B_(z) ^(res), given by Eq. (31),ensures E₁−E₀=ω_(y). In FIG. 18B shows the same interference effect seenin FIG. 17 . FIG. 18C shows the vacuum Rabi splitting. As indicated witha red arrow, the effective coupling, related to the separation betweenthe two peaks, corresponds to g_(s)/2π≈5 MHz and the parameters underconsideration can be readily achieved in Si DQD architectures.

FIG. 19 shows cavity transmission |A| as a function of Δ₀ close to theresonant field for different values of the charge-cavity couplingg_(c)/2π={40,80,160} MHz. The magnetic field has been slightly detunedfrom the resonance condition to make the relative heights of the vacuumRabi split modes the same. The rest of the parameters are t_(c)=15.4μeV, ϵ=0, B_(x)=1.62 μeV, γ_(c)/2π=100 MHz, g_(c)/2π=40 MHz, κ/2π=1.77MHz, and ω_(c)/2π=5.85 GHz≈24 μeV.

In the present discussion, a three-level system is at issue, where thespin-photon coupling is mediated by the spin-charge hybridization. Thethree-level system structure explains not only the interference but alsowhy the width and position of the two resonance peaks in FIG. 18C isslightly asymmetric. As expected, this asymmetry is more apparent asg_(c) increases, which is shown in FIG. 19 . As described furtherherein, the problem can be reduced to an equivalent two-level system tobe able to characterize the spin-photon coupling within the standardformalism utilized for the Jaynes-Cummings model.

The estimated value of the spin decoherence rate induced by thespin-charge hybridization is of the order γ_(s)/2π≈(1-10) MHz. Anothersource of decoherence in Si QDs is the effect of the Si nuclear spinswhich surround the electron spin. As their evolution is slow compared tothe typical time scale of the electronic processes, the nuclear spinseffectively produce a random magnetic field which slightly influencesthe total magnetic field on the DQD. This small perturbation of themagnetic field, B_(z) ^(tot)=B_(z)+B_(z) ^(nuc), will modify thefrequency ω_(c)+Δ₀ of the two vacuum Rabi normal modes, as can beextracted from FIG. 18A. The nuclear magnetic field follows a Gaussiandistribution with average zero and standard deviation σ^(nuc). In thisway, a Gaussian profile is superimposed to the Lorentzian profile of theresonances and the final width also depends on σ^(nuc). At the pointwith maximum spin-charge hybridization (Ω≈B_(z)), this effect isnegligible because the decoherence is dominated by charge decoherence.Away from this point, the two broadening mechanisms have to be combined,resulting in a Voigt profile. The spin dephasing times in natural Si are≈1 μs, which corresponds approximately to a standard deviation ofσ^(nuc)/2π≈0.3 MHz for the nuclear magnetic field distribution.According to Eq. (30), the positions of the two vacuum Rabi modes aregiven by the solutions of the equation−Δ₀ +g _(c) Re(χ₀₁ d ₀₁+χ₀₁ d ₀₂)=0,   (32)where the susceptibilities are a function of Δ₀ via the detuningsδ₁₍₂₎≡E₁₍₂₎−E₀−ω_(R) and ω_(R)=ω_(c)+Δ₀. As the magnetic field createdby the nuclear spins is small, one can expand the solutions Δ₀ ^(±)tofirst order to obtain

$\begin{matrix}\left. {\Delta_{0}^{\pm} \simeq {{\Delta_{0}^{\pm}\left( {B_{z}^{nuc} = 0} \right)} + \frac{\partial\Delta_{0}^{\pm}}{\partial B_{z}}}} \middle| {}_{B_{z}^{nuc} = 0}{B_{z}^{nuc}.} \right. & (33)\end{matrix}$Therefore, the broadening of the vacuum Rabi modes due to the nuclearspins is given by σ=|∂Δ₀ ^(±)/∂B_(z)|σ^(nuc) and the total spindecoherence rate isγ_(s) ^(tot)=γ_(s)/2+√{square root over ((γ_(s)/2)²+8(ln 2)σ²)}.   (34)The long spin dephasing times in Si allow the strong-coupling regime tobe reached approximately at the same working points. For instance, for atunnel coupling t_(c)≈15 μeV, the estimated spin dephasing rate inducedby charge hybridization is γ_(s)/2π≈2 MHz and the broadening due to thenuclear spins is given by σ/2π≈0.14 MHz; therefore, γ_(s) ^(tot)/2π≈2MHz.

F. Two-level equivalent system. To reduce the problem to a two-levelsystem, it is more convenient to work in the orbital basis. Using therelations in Eqs. (A8) and (A9) the QLEs can be rewritten in terms ofthe operators a, σ_(τ)=|−, ↓

+, ↑| and σ_(s)=|−, ↓

−, ↑|. Neglecting input noise terms for the charge relaxation which willbe irrelevant for our linear response theory, these equations read

$\begin{matrix}{{\overset{.}{a} = {{i\Delta_{0}a} - {\frac{\kappa}{2}a} + {\sqrt{\kappa_{1}}a_{{in},1}} + {{ig}_{c}\cos{\theta\sigma}_{\tau}}}},{{\overset{.}{\sigma}}_{t} = {{{- i}\Delta_{\tau}\sigma_{\tau}} - {\gamma_{c}\sigma_{\tau}} + {{ig}_{c}\cos\theta a}}}} & (35)\end{matrix}$ $\begin{matrix}{{{+ i}\frac{B_{x}\cos\theta}{2}\sigma_{s}},} & (36)\end{matrix}$ $\begin{matrix}{{{\overset{.}{\sigma}}_{x} = {{{- i}\Delta_{s}\sigma_{s}} + {i\frac{B_{x}\cos\theta}{2}\sigma_{\tau}}}},} & (37)\end{matrix}$where

$\Delta_{\tau(s)} = {{\pm \frac{\Omega - B_{z}}{2}} - E_{0} - {\omega_{R}.}}$As evident from these equations, although the electric field of thecavity only couples to the charge excitation, the spin-chargehybridization generates an effective spin-photon coupling.

Solving Eq. (36) for the steady state one can obtain the bright (B) modethat mediates this coupling,

$\begin{matrix}{{\sigma_{B} = {{\frac{{- 2}\cos{\theta\left( {\Delta_{\tau} - {i\gamma_{c}}} \right)}}{\sqrt{{4g_{c}^{2}} + B_{x}^{2}}}{\overset{\_}{\sigma}}_{\tau}} = {{\sin{\alpha\sigma}_{1}} + {\cos\alpha a}}}},} & (38)\end{matrix}$where is introduced the angle

$\alpha = {\arctan{\frac{B_{x}}{2g_{c}}.}}$For the Eqs. (35) and (37), the reduced dynamics include

$\begin{matrix}{{\overset{.}{a} = {{{i\left( {\Delta_{0} + {\Delta_{\tau}\eta\cos^{2}\alpha}} \right)}a} - {\frac{\kappa^{\prime}}{2}a} + {\sqrt{\kappa_{1}}a_{{in},1}} + {i\sin\alpha\cos{{\alpha\eta}\left( {\Delta_{\tau} + {i\gamma_{c}}} \right)}\sigma_{s}}}},} & (39)\end{matrix}$ $\begin{matrix}{{{{\overset{.}{\sigma}}_{s} = {{{- {i\left( {\Delta_{s} - {\Delta_{\tau}\eta\sin^{2}\alpha}} \right)}}\sigma_{s}} - {\gamma_{s}\sigma_{s}} + {i\sin\alpha\cos{{\alpha\eta}\left( {\Delta_{\tau} + {i\gamma_{c}}} \right)}a}}},{with}}{{\eta = {\frac{{B_{x}^{2}/4} + g_{c}^{2}}{\Delta_{\tau}^{2} + \gamma_{c}^{2}} + {\cos^{2}\theta}}},}} & (40)\end{matrix}$

and effective decay rates

$\begin{matrix}{{\kappa^{\prime} = {{\kappa + {2\gamma_{c}{\eta cos}^{2}\alpha}} = {\kappa + {2\gamma_{c}\frac{g_{c}^{2}\cos^{2}\theta}{\Delta_{\tau}^{2} + \gamma_{c}^{2}}}}}},} & (42)\end{matrix}$ $\begin{matrix}{\gamma_{s} = {{y_{c}{\eta sin}^{2}\alpha} = {\frac{\gamma_{c}}{4}{\frac{B_{\chi}^{2}\cos^{2}\theta}{\Delta_{\tau}^{2} + \gamma_{c}^{2}}.}}}} & (43)\end{matrix}$

According to the derivation in the “CHARACTERIZATION OF THE SPIN-PHOTONCOUPLING” section below, the resonance condition reads

$\begin{matrix}{{\left( {\Delta_{s} + \Delta_{0}} \right)^{res} = {{- \Delta_{\tau}}\eta\frac{{2\gamma_{c}\eta} + {\kappa\cos 2\alpha}}{\kappa + {2\gamma_{c}{\eta cos}2\alpha}}}},} & (44)\end{matrix}$

and strong coupling is achieved for

$\begin{matrix}{{{\mathcal{g}}_{s} > \Gamma \equiv \frac{❘{\kappa + {2\gamma_{c}{\eta cos2\alpha}}}❘}{2\sqrt{2}\sqrt{\frac{{4\gamma_{c}^{2}\eta^{2}} + {4\gamma_{c}{\eta\kappa cos2\alpha}} + \kappa^{2}}{{4\gamma_{c}^{2}\eta^{2}} + {4\gamma_{c}{\eta\kappa cos}^{2}\alpha} + \kappa^{2}}}}},} & (45)\end{matrix}$

where g_(s) is defined as

$\begin{matrix}{g_{s} = {{{❘\Delta_{\tau}❘}{\eta sin\alpha cos\alpha}} = {{❘\Delta_{\tau}❘}{\frac{B_{\chi}g_{c}\cos^{2}\theta}{2\left( {\Delta_{\tau}^{2} + \gamma_{c}^{2}} \right)}.}}}} & (46)\end{matrix}$

FIG. 20 shows strength of the spin-cavity coupling

_(s)/Γ according to Eq. (45) as a function of t_(c) and ϵ. B_(z) hasbeen adjusted to the resonance condition Eq. (44). Note that thestrong-coupling regime (g_(s)/Γ>1) is achieved away from the blackdashed line, Ω=ω_(c)=24 μeV, in agreement with FIG. 16B. The otherparameters are B_(χ)=1.62 μeV, γ_(c)/2π=(100+150|sin θ|) MHz,g_(c)/2π=40 MHz, and κ/2π=1.77 MHz.

In FIG. 20 , a map is shown of the coupling strength via the quantityg_(s)/Γ, with the magnetic field adjusted to the resonance condition.This map indicates the optimum working points that create a strongspin-photon interaction that overcomes the decoherence.

In conclusion, the conditions for achieving strong coupling between asingle electron spin and a microwave cavity photon are shown, whichallow long distance spin-spin coupling and long-range spin-qubit gates.Nonlocal quantum gates may also facilitate quantum error correctionwithin a fault-tolerant architecture.

The analysis on the dynamics of the full hybrid siliconcQED systemconfirms that, with the recent advances in Si DQDs fabrication andcontrol, a spin-photon coupling of more than 10 MHz with a sufficientlylow spin decoherence rate is achievable with this setup, potentiallyallowing the strong-coupling regime. In such a regime, the cavity notonly can act as a mediator of spin-spin coupling but also enablescavity-based readout of the spin qubit state. Interestingly, thestrong-coupling regime for the spin-cavity coupling may be attained evenwhen the coupling strength of the charge-cavity coupling cannot overcomethe charge decoherence rate.

Although here the discussion in this section has focused on the couplingof a single electron spin to a single photon, the implementation ofproposals for other types of spin qubits with more than one electron isfeasible with the present technology in Si QDs.

Decoherence Model

It is assumed that the charge relaxation processes dominates over directspin relaxation. The most relevant sources of decoherence in the presentsystem are charge relaxation effects due to the phonon environment (γ₁)and dephasing due to charge noise (γ_(ϕ)); therefore, the Liouvilliancan be written as

$\begin{matrix}{{{\mathcal{L}_{ph}\rho} = {{\frac{\gamma_{1}}{2}\left( {{2\sigma_{-}{\rho\sigma}_{+}} - {\sigma_{+}\sigma_{-}\rho} - {{\rho\sigma}_{+}\sigma_{-}}} \right)} + {\frac{\gamma\phi}{4}\left( {{2\sigma_{z}{\rho\sigma}_{z}} - {\sigma_{z}\sigma_{z}\rho} - {{\rho\sigma}_{z}\sigma_{z}}} \right)}}},} & ({A1})\end{matrix}$

where σ_(±)=|±

∓| and σ_(z)=|+

+|−|−

−|. (Note that in this section the Pauli operators σα are in the basisof bonding and antibonding states, |±

, instead of left and right.)

The interaction Hamiltonian for charge decays can be written as

$\begin{matrix}{{{Hph} = {\sum\limits_{k}{c_{k}\left( {{\left. ❘ + \right\rangle\left\langle - ❘ \right.b_{k}} + {\left. ❘ - \right\rangle\left\langle + ❘ \right.b_{k}}} \right)}}},} & ({A2})\end{matrix}$

where b_(k) annihilates a phonon in mode k. Therefore, the relaxationrate at zero temperature can be obtained using Fermi's golden rule,

$\begin{matrix}{{\gamma_{1} = {\frac{2\pi}{\hslash}{\sum\limits_{f}{{❘{\left\langle f❘ \right.\left\langle - ❘ \right.H_{ph}\left. ❘ + \right\rangle\left. ❘0 \right\rangle}❘}^{2}{\delta\left( {\Omega - E_{f}} \right)}}}}},} & ({A3})\end{matrix}$

where |0

and |f

are the initial phonon vacuum and single-phonon final states, Ω is theorbital energy (Ω=√{square root over (ϵ²+4t_(c) ²)}), and E_(f) denotesthe phonon energy. Substituting Eq. (A2) into Eq. (A3), the result is

$\begin{matrix}{{\gamma_{1} = {{\frac{2\pi}{\hslash}{\sum\limits_{k}{{❘c_{k}❘}^{2}{\delta\left( {\Omega - E_{k}} \right)}}}} = {\frac{2\pi}{\hslash}{❘c_{k}❘}^{2}{D(\Omega)}}}},} & ({A4})\end{matrix}$

where k is the modulus of the k vector evaluated at the energy Ω. Here,D(E) is the phonon density of states. In general, γ₁ depends on theparameters t_(c) and ϵ both via D(Ω) and c_(k), since k=k(Ω). A constantγ₁ is assumed since it is expected that this approximation will hold ina small transition energy window around the cavity frequency.

The pure dephasing term is due to charge noise in the environment. Acontribution to the dephasing rate γ_(ϕ) is proportional to the firstderivative of the energy transition with respect to ϵ, i.e.,

$\begin{matrix}{{{\gamma_{\phi}^{(1)} \sim \frac{\partial\left( {E_{+} - E_{-}} \right)}{\partial\epsilon}} = {\sin\theta}},} & ({A5})\end{matrix}$

and therefore is zero at the “sweet spot” ϵ=0. However, as observed inrecent experiments, the charge noise at the sweet spot cannot beneglected. To account for this observation, an offset value to γ_(ϕ) hasbeen added, which is modeled then as γ_(ϕ)=γ_(ϕ) ⁽⁰⁾+γ_(ϕ) ⁽¹⁾.

Using the Liouvillian in Eq. (A1), one can calculate the decoherencedynamics for the mean value of any operator as

{dot over (A)}

=tr{A

_(ph)ρ}. (In the following the brackets are omitted for simplicity.) Thecoherences decay as

$\begin{matrix}{\overset{.}{\sigma_{\pm}} = {{{- \gamma_{c}}\sigma_{\pm}} = {{- \left( {\frac{\gamma_{1}}{2} + \gamma_{\phi}} \right)}{\sigma_{\pm}.}}}} & ({A6})\end{matrix}$

In this discussion, the spin degree of freedom is included andspin-independent rates are assumed. In the main discussion above, it wasdefined σ_(τ)=|−, ↓

+, ↓| and σ_(s)=|−, ↓

+, ↑|. While σ_(τ) decays as {dot over (σ)}_(τ)=−γ_(c)σ_(r), the sametype of calculation taking into account the spin reveals {dot over(σ)}_(s)=0. The decoherence part of the dynamics entering in Eq. (5) isobtained via a rotation of the previous uncoupled equations into theeigenbasis of H₀, which results in

$\begin{matrix}{{\overset{.}{\sigma}}_{nm}{\sum\limits_{n^{\prime},m^{\prime}}{\gamma_{{nm},{n^{\prime}m^{\prime 0}n^{\prime}m^{\prime}}}.}}} & ({A7})\end{matrix}$

From the form of the eigenstates in the bonding-antibonding basis [Eqs.(13) to (16)], one can determine the effect of charge dephasing in theeigenbasis of H₀. Since

$\begin{matrix}{{\sigma_{01} \simeq {{\cos\frac{\Phi}{2}\sigma_{s}} + {\sin\frac{\Phi}{2}\sigma_{\tau}}}},} & ({A8})\end{matrix}$ $\begin{matrix}{{\sigma_{02} \simeq {{\sin\frac{\Phi}{2}\sigma_{s}} - {\cos\frac{\Phi}{2}\sigma_{\tau}}}},} & ({A9})\end{matrix}$

the decoherence dynamics can be expressed as

$\begin{matrix}{\begin{pmatrix}{\overset{.}{\sigma}}_{01} \\{\overset{.}{\sigma}}_{02}\end{pmatrix} \simeq {{- {\gamma_{c}\left( {\begin{matrix}{\sin^{2}\frac{\Phi}{2}} \\{- \frac{\sin\Phi}{2}}\end{matrix}\begin{matrix}{- \frac{\sin\Phi}{2}} \\{\cos^{2}\frac{\Phi}{2}}\end{matrix}} \right)}}{\begin{pmatrix}\sigma_{01} \\\sigma_{02}\end{pmatrix}.}}} & ({A10})\end{matrix}$

Note also that σ₀₃≃|−, ↓

+, ↑|; therefore, its decoherence is decoupled from the othertransitions, {dot over (σ)}₀₃≃(γ₁+γ_(ϕ))σ₀₃/2.

Multilevel RWA

The time-dependent equations of motion, Eqs. (4) and (5), can be solvedwithin a rotating-wave-approximation (RWA) if the driving frequency isclose to the transition energies of the system. Defining {tilde over(σ)}_(n,n+j)=σ_(n,n+j)e^(iω) ^(R) ^(t) for j>0, these equations includeboth time-independent terms and terms which oscillate at frequency2ω_(R),

$\begin{matrix}{{\overset{.}{a} = {{i\Delta_{0}a} - {\frac{\kappa}{2}a} + {\sqrt{\kappa_{1}}a_{{in},1}} + {\sqrt{\kappa_{2}}a_{{in},2}} - {{ig}_{c}{\sum\limits_{n = 0}^{2}{\sum\limits_{j = 1}^{3 - n}{d_{n,{n + j}}{\overset{\sim}{\sigma}}_{n,{n + j}}}}}} - {{ig}_{c}{\sum\limits_{n = 1}^{3}{\sum\limits_{j = 1}^{n}{d_{n,{n - i}}{\overset{\sim}{\sigma}}_{n,{n - j}}e^{2i\omega_{R}t}}}}}}},} & ({B1})\end{matrix}$ $\begin{matrix}{{\overset{.}{\overset{\sim}{\sigma}}}_{n,{n + j}} = {{{- {i\left( {E_{n + j} - E_{n} - \sigma_{R}} \right)}}{\overset{\sim}{\sigma}}_{n,{n + j}}} - {\sum\limits_{n^{\prime}j^{\prime}}{\gamma_{a,{n + I},n,{n^{\prime} + i}}{\overset{\sim}{\sigma}}_{n^{\prime},{n^{\prime} + j^{\prime}}}}} - {\sum\limits_{n^{\prime},j^{\prime}}{\gamma_{n,{n + j},n^{\prime},{n^{\prime} - j^{\prime}}}e^{2i\omega_{R}t}{\overset{\sim}{\sigma}}_{R^{\prime},{n^{\prime} - j^{\prime}}}}} + {\sqrt{2\gamma}\mathcal{F}e^{i\omega{gt}}} - {{{ig}_{c}\left( {a + {a^{\dagger}e^{2i\omega{gt}}}} \right)}d_{{n + j},n}{\delta_{n,0}.}}}} & ({B2})\end{matrix}$

Here, j,j′>0. The RWA consists in neglecting the fastoscillating, i.e.,counter-rotating terms. For the mean value of the operators and usingσ_(n,n+j) instead of {tilde over (σ)}_(n,n+j) to simplify the notation,the equations in the RWA read

$\begin{matrix}{{\overset{.}{a} = {{i\Delta_{0}a} - {\frac{\kappa}{2}a} + {\sqrt{\kappa_{1}}a_{{in},1}} - {{ig}_{c}{\sum\limits_{n = 0}^{2}{\sum\limits_{j = 1}^{3 - n}{d_{n,{n + j}}\sigma_{n,{n + j}}}}}}}},} & ({B3})\end{matrix}$ $\begin{matrix}{{{\overset{.}{\sigma}}_{n,{n + j}} = {{{- {i\left( {E_{n + j} - E_{n} - \omega_{R}} \right)}}\sigma_{n,{n + j}}} - {\sum\limits_{n^{\prime}j^{\prime}}{\gamma_{n,{n + j},n^{\prime},{n^{\prime} + j^{\prime}}}\sigma_{n^{\prime},{n^{\prime} + j^{\prime}}}}} - {{ig}_{c}{ad}_{{n + j},n}\delta_{a0}}}},} & ({B4})\end{matrix}$

since

a_(in,2)

=0 and

=0. In the equations, as in the main discussion above, the brackets areomitted for simplicity.

General Magnetic Field Gradient Direction

In the discussion above, it has been assumed that the micromagnet in agiven external magnetic field introduces a gradient of the magneticfield in a perpendicular direction between the positions of the twoquantum dots. In a more realistic situation, the micromagnet could alsointroduce a net magnetic field in this perpendicular direction or agradient in the direction of the external magnetic field. Thesesituations can be described in a general way with a model Hamiltonianlike the one presented in the discussion above [Eq. (1)] plus acontribution corresponding to a magnetic field gradient in z directionbetween the two quantum dots, i.e., H₀′=H₀+Δ_(z)σ_(z)τ_(z)/2.

Analogous to Eq. (8), in the product basis of antibonding and bondingorbitals ± with spin ↑↓ in the z direction, {|+, ↑

, |−, ↑

, |+, ↓

, |−, ↓

}, the Hamiltonian reads

$\left( {C1} \right){{H_{0}^{\prime{orb}} = {\frac{1}{2}\begin{pmatrix}{{\Omega_{+}\cos\alpha_{+}} + B_{z}} & {{- \Omega_{+}}\sin\alpha_{+}} & {B_{x}\sin\theta} & {{- B_{x}}\cos\theta} \\{{- \Omega_{+}}\sin\alpha_{+}} & {{{- \Omega_{+}}\cos\alpha_{+}} + B_{z}} & {{- B_{x}}\cos\theta} & {{- B_{x}}\sin\theta} \\{B_{x}\sin\theta} & {{- B_{x}}\cos\theta} & {{\Omega_{-}\cos\alpha_{-}} - B_{z}} & {\Omega_{-}\sin\alpha_{-}} \\{{- B_{x}}\cos\theta} & {{- B_{x}}\sin\theta} & {\Omega_{-}\sin\alpha_{-}} & {{{- \Omega_{-}}\cos\alpha_{-}} - B_{z}}\end{pmatrix}}},}$where it is defined the two different orbital energies Ω_(±)==√{squareroot over (4t_(c) ²(ϵ±Δ_(z))²)} and the angles

$\begin{matrix}{\alpha_{\pm} = {\arctan{\frac{2t_{c}\Delta_{z}}{{4t_{c}^{2}} + {\epsilon\left( {\epsilon \pm \Delta_{z}} \right)}}.}}} & ({C2})\end{matrix}$

As expected, the parallel gradient Δ_(z) couples bonding and antibondingstates with the same spin. Now, a rotation can be applied to the orbitalstates from {|+, σ

, |−, σ

} to the new {|+′, σ

, |−′, σ

} such that

$\begin{matrix}{H_{0}^{\prime{orb}} = {\frac{1}{2}{\begin{pmatrix}{\Omega_{+} + B_{z}} & 0 & {B_{x}{\sin\left( {\theta + \frac{\alpha_{+} - \alpha_{-}}{2}} \right)}} & {{- B_{x}}{\cos\left( {\theta + \frac{\alpha_{+} - \alpha_{-}}{2}} \right)}} \\0 & {{- \Omega_{+}} + B_{z}} & {{- B_{x}}{\cos\left( {\theta + \frac{\alpha_{+} - \alpha_{-}}{2}} \right)}} & {{- B_{x}}{\sin\left( {\theta + \frac{\alpha_{+} - \alpha_{-}}{2}} \right)}} \\{B_{x}{\sin\left( {\theta + \frac{\alpha_{+} - \alpha_{-}}{2}} \right)}} & {{- B_{x}}{\cos\left( {\theta + \frac{\alpha_{+} - \alpha_{-}}{2}} \right)}} & {\Omega_{-} - B_{z}} & 0 \\{{- B_{x}}{\cos\left( {\theta + \frac{\alpha_{+} - \alpha_{-}}{2}} \right)}} & {{- B_{x}}{\sin\left( {\theta + \frac{\alpha_{+} - \alpha_{-}}{2}} \right)}} & 0 & {{- \Omega_{-}} - B_{z}}\end{pmatrix}.}}} & ({C3})\end{matrix}$

Therefore, under approximately the same conditions as in the case of thediscussion above, the eigenstates can be approximated by

$\begin{matrix}{{\left. ❘1 \right\rangle \simeq {{\cos\frac{\Phi^{\prime}}{2}\left. ❘{-^{\prime}\left. ,\uparrow \right.} \right\rangle} + {\sin\frac{\Phi^{\prime}}{2}\left. ❘{+^{\prime}\left. ,\downarrow \right.} \right)}}},} & ({C4})\end{matrix}$ $\begin{matrix}{{\left. ❘2 \right\rangle \simeq {{\sin\frac{\Phi^{\prime}}{2}\left. ❘{-^{\prime}\left. ,\uparrow \right.} \right\rangle} - {\cos\frac{\Phi^{\prime}}{2}\left. ❘{+^{\prime}\left. ,\downarrow \right.} \right)}}},} & ({C5})\end{matrix}$ $\begin{matrix}{{\left. ❘0 \right\rangle \simeq {\sin\frac{\Phi}{2}\left. ❘{-^{\prime}\left. ,\downarrow \right.} \right)}},} & ({C6})\end{matrix}$ $\begin{matrix}{{\left. ❘3 \right\rangle \simeq \left. ❘{+^{\prime}\left. ,\uparrow \right.} \right)},} & ({C7})\end{matrix}$with the spin-orbit mixing angle

$\begin{matrix}{{\Phi^{\prime} = {\arctan\frac{B_{x}{\cos\left( {\theta + \frac{\alpha_{+} - \alpha_{-}}{2}} \right)}}{\frac{\Omega_{+} + \Omega_{-}}{2} - B_{z}}}},} & ({C8})\end{matrix}$

Finally, the relevant dipole matrix elements read

$\begin{matrix}{{{d_{01} \simeq} - {{\cos\left( {\theta - \alpha_{-}} \right)}\sin\frac{\Phi^{\prime}}{2}}},} & ({C9})\end{matrix}$ $\begin{matrix}{d_{02} \simeq {\cos\left( {\theta - \alpha_{-}} \right)}\cos{\frac{\Phi^{\prime}}{2}.}} & ({C10})\end{matrix}$

In FIGS. 21A-B, the expected spin-photon coupling strength isrepresented as a function of the DQD detuning and the tunnel coupling.As compared to FIG. 16A, the map in FIG. 21A is distorted due to theintroduced asymmetry, but importantly a strong coupling can be alsoachieved, even for a large value of Δ_(z).

Characterization of the Spin-Photon Coupling

The calculation of the strong-coupling condition for a two-levelequivalent system is described in this section. For this, first isanalyzed the simpler case of a Jaynes-Cummings model, often studied inthe literature. This model describes the interaction of a two-levelsystem (TLS) to a photonic mode in a cavity in the RWA. The Hamiltonianreads H=−Δ₀a^(†)a+Δ_(JC)σ_(z)−g_(JC)(a^(†)σ_+aσ⁺), where Δ₀ is thedetuning between the driving and the cavity and Δ_(JC) the detuningbetween the TLS and the driving. The TLS decoherence rate is γ_(JC) andthe cavity decay rate is K. If the transition energy of the two-levelsystem is near the cavity frequency, one can probe the coherentlight-matter interaction by driving the cavity close to resonance andobserving resonances coming from the hybridized one-excitationlight-matter states, superpositions of the excited states {|0,n+1

, |1, n

}, where n is the number of photons in the cavity and 0(1) the two-levelstate. However, this is only observable if the coupling is strong enoughin comparison with the decay rates. To determine how strong the couplingshould be in order to observe separate resonances, it is useful todiagonalize the non-Hermitian Hamiltonian in the one-excitation subspacecontaining the decay rates,

$\begin{matrix}{H_{JC} = {- {\begin{pmatrix}{\Delta_{0} + {i{\kappa/2}}} & g_{JC} \\g_{JC} & {{- \Delta_{JC}} + {i\gamma_{JC}}}\end{pmatrix}.}}} & ({D1})\end{matrix}$

The eigenvalues are of the formμ_(±) =A−iB/2±√{square root over (C+iD,)}   (D2)with A, B, C, D all real. A can be seen to be the average frequency ofthe system, while B is the average damping. However, the C and D termsare more subtle. The strong-coupling condition is defined as having asufficiently large interaction such that two separated peaks areobservable in the photon response (C>B²/4) and system modes that arenear equal combinations of matter and light (C>0, D=0), which occurswhen Δ₀=−Δ_(JC). If this is the case, the combined condition is

$\begin{matrix}{g_{JC} > \sqrt{\frac{\gamma_{JC}^{2} + \left( {\kappa/2} \right)^{2}}{2},}} & ({D3})\end{matrix}$which reduces to the usual g_(JC)>γ_(JC), K/2 for 2_(γ) _(JC) ≈K.

FIG. 21A shows expected effective coupling g_(s)/g_(c)=|d₀₁₍₂₎|,according to Eqs. (C9) and (C10) as a function of tc and ϵ. The blackdashed line corresponds to Ω₊+Ω_=2B_(z) and separates the region whereg_(s)/g_(c)=|d_(0,1)| (above) from the region whereg_(s)/g_(c)=|d_(0,2)| (below). FIG. 21B shows vertical cuts for ϵ=0(black) and ϵ=20 μeV (blue) for different values of Δ_(z). Chosen valuesinclude B_(x), 1.62 μeV, B_(z)=24 μeV and the values of Δ_(z) indicatedon the figures.

Let us now consider weak versus strong coupling for the two-levelequivalent system, whose interaction is described by Eqs. (39) and (40).In the one-excitation subspace, the effective non-Hermitian Hamiltonianreads

$\begin{matrix}{H_{eff} = {{- \begin{pmatrix}{\Delta_{0} + {i{\kappa^{\prime}/2}}} & 0 \\0 & {{- \Delta_{s}} + {i\gamma_{s}}}\end{pmatrix}} - {{\eta\begin{pmatrix}{\Delta_{\tau}\cos^{2}\alpha} & {\left( {\Delta_{\tau} + {i\gamma_{c}}} \right)\sin\alpha\cos\alpha} \\{\left( {\Delta_{\tau} + {i\gamma_{c}}} \right)\sin{\alpha cos}\alpha} & {\Delta_{\tau}\sin^{2}\alpha}\end{pmatrix}}.}}} & ({D4})\end{matrix}$

The eigenvalues of this Hamiltonian have the same form; therefore,resonance can be defined as D=0 and strong coupling as C>B²/4, as above.First is examined the on-resonance condition. The detuning for resonanceis

$\begin{matrix}{\left( {\Delta_{s} + \Delta_{0}} \right)^{res} = {{- \Delta_{\tau}}\eta{\frac{{2\gamma_{c}\eta} + {\kappa\cos 2\alpha}}{\kappa + {2\gamma_{c}\eta\cos 2a}}.}}} & ({D5})\end{matrix}$

This value corresponds to setting the detuning to match the (quantum)“Stark shifted” response of the spin and the photon. If one does notchoose the detunings such that D=0, then the two vacuum Rabi-split peakswill have different linewidths. A modification of the strong-couplingcondition is also found. This arises for

$\begin{matrix}{g_{s} > \Gamma \equiv \frac{❘{\kappa + {2\gamma_{c}{\eta cos2\alpha}}}❘}{2\sqrt{2}\sqrt{\frac{{4\gamma_{c}^{2}\eta^{2}} + {4\gamma_{c}\eta{\kappa cos}2\alpha} + \kappa^{2}}{{4\gamma_{c}^{2}\eta^{2}} + {4\gamma_{c}\eta{\kappa cos}^{2}\alpha} + \kappa^{2}}}}} & ({D6})\end{matrix}$with the definition

$\begin{matrix}{g_{s} = {{{❘\Delta_{\tau}❘}\eta\sin{\alpha cos}\alpha} = {{❘\Delta_{\tau}❘}{\frac{B_{x}g_{c}\cos^{2}\theta}{2\left( {\Delta_{\tau}^{2} + \gamma_{c}^{2}} \right)}.}}}} & ({D7})\end{matrix}$Third Section: Long-Range Microwave Mediated Interactions BetweenElectron Spins

Nonlocal qubit interactions are a hallmark of advanced quantuminformation technologies. The ability to transfer quantum states andgenerate entanglement over distances much larger than qubit lengthscales greatly increases connectivity and is an important step towardsmaximal parallelism and the implementation of two-qubit gates onarbitrary pairs of qubits. Qubit coupling schemes based on cavityquantum electrodynamics also offer the possibility of using high qualityfactor resonators as quantum memories. Extending qubit interactionsbeyond the nearest neighbor is particularly beneficial for spin-basedquantum computing architectures, which are limited by short-rangeexchange interactions. Despite rapidly maturing device technology forsilicon spin qubits, experimental progress towards achieving long-rangespin-spin coupling has so far been restricted to interactions betweenindividual spins and microwave photons. Here is demonstratedmicrowave-mediated spin-spin coupling between two electrons that arephysically separated by more than 4 mm. An enhanced vacuum Rabisplitting is observed when both spins are tuned into resonance with thecavity, indicative of a coherent spin-spin interaction. The resultsdemonstrate that microwave-frequency photons can be used as a resourceto generate long-range two-qubit gates between spatially separatedspins.

The devices studied in this example comprise (e.g., or consist of) twodouble quantum dots (DQDs), denoted L-DQD and R-DQD, that are fabricatedon a Si/SiGe heterostructure and positioned at the antinodes of ahalf-wavelength Nb superconducting cavity (FIG. 22A). Device 1 isfabricated on a nat-Si quantum well and has a cavity center frequencyf_(c)=6.745 GHz and decay rate κ/2π=1.98 MHz. Device 2 utilizes anenriched Si quantum well with an 800 ppm residual concentration of Si. Asingle electron is isolated in each DQD and interacts with the electricfield of the cavity through the electric dipole interaction. The devicedesign uses a split-gate cavity coupler (labelled CP in FIG. 22B) thatis galvanically connected to the center pin of the superconductingcavity.

First is demonstrated a strong coupling of a spin trapped in each DQD toa cavity photon. In the example device architecture a large electricdipole coupling rate g_(c)/2π≈40 MHz is combined with an artificialspin-orbit interaction generated by a micromagnet to achieve spin-photoncoupling. To probe spin-photon coupling, the cavity transmission A/A₀ isplotted as a function of cavity probe frequency f and B^(ext) in FIG.22C. The Zeeman splitting increases with the total magnetic field {rightarrow over (B)}^(tot)={right arrow over (B)}^(ext)+{right arrow over(B)}^(M), where {right arrow over (B)}^(ext) is the externally appliedfield and {right arrow over (B)}^(M) is the stray field of themicromagnet. Field tuning allows bringing each spin into resonance withthe cavity photon of energy hf_(c), where h is Planck's constant.Coherent coupling between the spin trapped in the L-DQD (L-spin) and thecavity photon is observed, as evidenced by the vacuum Rabi splitting,when B^(ext)=109.1 mT. Strong spinphoton coupling is achieved with thespin-cavity coupling rate g_(s,L)/2π=10.7±0.1 MHz exceeding the cavitydecay rate κ/2π=1.98 MHz and spin decoherence rate γ_(s,L)/2π=4.7 MHz.Likewise, strong spin-photon coupling is observed for the R-spin atB^(ext)=103.1 mT with g_(s,L)/2π=12.0±0.2 MHz exceeding γ_(s,L)/2π=5.3MHz and κ/2π. Factoring in the susceptibility of the micromagnet, χ≈0.6,the 6 mT difference in B^(ext) is equivalent to a 268 MHz difference inthe spin resonance frequencies and precludes the observation of resonantspin-spin coupling.

FIG. 22A-D shows an example cavity-coupler for spins. FIG. 22A shows anexample optical micrograph of the superconducting cavity containing twosingle electron DQDs. The electron spin in each DQD is coupled to thecavity through a combination of electric-dipole and artificialspin-orbit interactions. FIG. 22B shows an example false-color scanningelectron microscope image of the L-DQD. A double well potential isformed beneath plunger gates P1L and P2L, and the barrier gate B2L isused to adjust interdot tunnel coupling. Spin-orbit coupling is inducedby a Co micromagnet (dashed lines). FIG. 22C shows cavity transmissionA/A₀ as a function of B^(ext) and ƒ. Vacuum Rabi splitting, a hallmarkof strong coupling, is observed for each spin. For a field angle ϕ=0°the R-spin and L-spin resonance conditions are separated by B^(ext)=6mT. FIG. 22D shows that increasing the field angle to ϕ=2.8°dramatically reduces the field separation.

To compensate for local differences in the magnetic field, the devicewas purposely fabricated with the long axis of the Co micromagnetstilted by ϕ=±15 degrees relative to the interdot axis of the DQDs (e.g.,FIG. 22B). Because of this intentional asymmetry, adjusting the angle ϕof the in-plane magnetic field relative to the DQD axis provides anadditional degree of freedom for simultaneous tuning of both spins intoresonance with the cavity. Qualitatively, the high permeability Comicromagnet concentrates the magnetic field lines, leading to a maximumtotal field when {right arrow over (B)}^(ext) is aligned with the longaxis of the micromagnet. This technique is well-suited for controllingthe spin resonance frequencies in FIG. 1 d , where the magnetic fieldangle is set to ϕ=2.8°. Here, the R-spin resonance condition shifts upto B^(ext)=104.7 mT and the L-spin resonance condition shifts down toB^(ext)=107.4 mT.

FIG. 23A shows expected spin resonance frequencies as a function of ϕfor B^(ext)=106.3 mT (top panel) and B^(ext)=110 mT (bottom panel).B^(ext) allows control over both spin frequencies with respect to thecavity, while 0.0 allows control over the spin frequencies with respectto each other. The frequency of the left (right) spin is plotted in blue(purple). FIG. 23B shows A/A₀ as a function of ƒ and ϕ demonstratessimultaneous tuning of both spins into resonance with the cavity atϕ=5.6° and B^(ext)=106.3 mT. Dashed lines indicate left and right spintransition frequencies. FIG. 23C shows an example theoretical predictionfor A/A₀.

Control over the difference between the spin resonance frequencies isdemonstrated by fixing the external magnetic field magnitude B^(ext) andvarying the angle ϕ of B^(ext) in the plane of the sample. The expectedspin resonance frequencies are plotted in FIG. 23A as a function of ϕwith B^(ext)=106.3 mT (upper panel) and B^(ext)=110 mT (lower panel),confirming that these two control parameters will bring the two spinsinto resonance with the cavity and each other. Based on microwavespectroscopy measurements of the spins, resonance is expected to occuraround ϕ=6° and B^(ext)=106.3 mT. FIG. 23B maps out the field angledependence of the spin resonance frequencies over a range ϕ=3-8° withB^(ext)=106.3 mT. The resonance frequency of the R-spin monotonicallymoves to lower f as ϕ is increased, while the L-spin shows the oppositedependence. With ϕ=5.6° both spins are tuned into resonance with thecavity. These results are well captured by the theoretical prediction oftransmission through the cavity shown in FIG. 23C.

The spectrum of the Jaynes-Cummings model for a single spin and a singlephoton in the cavity is shown in FIG. 24A. With the L-spin tuned intoresonance with the cavity, the spin and cavity photon hybridize leadingto a vacuum Rabi splitting of magnitude 2g_(s,L) in the cavitytransmission. In contrast, when both spins are simultaneously tuned intoresonance with the cavity, the excited state spectrum of theJaynes-Cummings model splits into a sub-radiant state and two brightstates. For N identical spins the Jaynes-Cummings model predicts a√{square root over (N)} enhancement of the coupling rate. In our devicegeometry, the sub-radiant state is the spin-triplet |0, T₀

=1/√{square root over (2)}(|0, ↑, ↓

+|0, ↑, ↓

) because the DQDs are located at opposite ends of the cavity where theelectric fields are 180 out of phase. Here, the spin states of theL/R-spin are quantized along their local total magnetic field axis. Thebright states are hybridizations between the singlet state |0, S₀

=1/√{square root over (2)}(|0, ↑, ↓

+|0, ↑, ↓

) and the state with a single photon |1, ↓, ↓

that are separated in energy by twice the collectively enhanced vacuumRabi coupling, 2g_(s,LR)=2√{square root over (g_(s,L) ²+g_(s,R) ²)}.Evidence of cavity-mediated single spin coupling is further discussedbelow.

FIG. 24B shows A/A₀ as a function of ƒ and B^(ext) with ϕ=5.6°, whereboth spins are in resonance with the cavity. As B^(ext) is increased,both spins are simultaneously tuned into resonance with the cavity atB^(ext)=106.3 mT resulting in observation of an enhancement in thevacuum Rabi splitting relative to the data sets shown in FIGS. 22C-D.The vacuum Rabi splitting is quantitatively analyzed for Device 1 inFIG. 24C and Device 2 in FIG. 24D. These devices have slight differencesin gate geometry, and the micromagnets in Device 2 are canted at anglesof θ=±10 degrees. In Device 1, g_(am) is extracted by measuring A/A₀ atϕ=5.6° and detuning the right/left spin using ϵ. In Device 2, theg_(s,L/R) is extracted by measuring A/A₀ at an off-resonant angle, atthe now separated resonant fields. For Device 1, a vacuum Rabi splittingof 2g_(s,L)/2π=21.4±0.2 MHz is observed when the L-spin is in resonancewith the cavity, 2g_(s,R)/2π=24.0±0.4 MHz when the R-spin is inresonance with the cavity, and 2g_(s,LR)/2π=30.2±0.2 MHz when both spinsare in resonance with the cavity. Device 2 shows similar behavior andagain exhibits an enhanced vacuum Rabi splitting 2g_(s,LR)/2π=18.4±0.4MHz when both spins are tuned into resonance with the cavity, comparedto the individual splittings 2g_(s,L)/2π=13.2±0.2 MHz and2g_(s,R)/2π=12.4±0.2 MHz. Combined, these two data sets give strongevidence for microwave assisted spin-spin interactions across a 4 mmlength scale that is many orders of magnitude larger than what can beachieved using direct wavefunction overlap. Moreover, these measurementsshow that both field angle and DQD level detuning can be used tomodulate the strength of the spin-spin interactions.

The data in FIGS. 24C-D are fit using a master equation description ofthe spin-cavity system. The Zeeman splittings ℏω_(L/R) and linewidthsγ_(s,L/R) are independent measured using microwave spectroscopy. Thecavity linewidth κ, as well as a complex Fano factor q, are obtained byfitting the bare cavity response with the spins detuned from resonance.The spin-photon coupling rates g_(s,L/R) for each device are obtained byfitting the data with the spins individually tuned into resonance withthe cavity as shown in FIGS. 24C-D. From the Jaynes-Cummings model thebright states are expected to split with the enhanced collectivecoupling rate 2g_(s,LR)=2√{square root over (g_(s,L) ²+g_(s,R) ²)}. Theextracted splittings agree with the theoretical predictions of2g_(s,LR)/2π=32.1 MHz for Device 1 and 2g_(s,LR)/2π=18.1 MHz for Device2 to within 6%.

FIGS. 24A-D show cavity-mediated spin-spin coupling. FIG. 24A showstuning the L-spin into resonance with the cavity results in vacuum Rabisplitting with magnitude 2g_(s,L). A vacuum Rabi splitting of magnitude2g_(s,LR)=2√{square root over (g_(s,L) ²+g_(s,R) ²)} is expected whenboth spins are tuned into resonance with the cavity. FIG. 24B shows A/A₀as a function of ƒ and B^(ext) with ϕ=5.6° indicates an enhanced vacuumRabi splitting when the L-spin and R-spin are tuned into resonance withthe cavity. FIG. 24C shows A/A₀ as a function of ƒ for the R-spin inresonance (upper curve), L-spin in resonance (middle curve), and bothspins in resonance (bottom curve). The enhancement of the vacuum Rabisplitting with both spins on resonance with the cavity is indicative ofspin-spin coupling via the cavity mode. FIG. 24D shows cavity-assistedspin-spin coupling is also observed in a second device with a differentgate pattern. Dashed lines in FIGS. 24C-D are fits to a master equationsimulation. Insets in FIGS. 24C-D are scanning electron microscopeimages of the devices, with 200 nm scale bars.

The observation of enhanced vacuum Rabi splitting when the separatedspins are simultaneously on resonance with the cavity is evidence oflong-range spin-spin coupling. The nonlocal interaction of two spinsmarks an important milestone for all-to-all qubit connectivity andscalability in silicon-based quantum circuits. In the near term, thisdemonstration shows that the disclosed techniques allow for theimplementation of modular qubit architectures in silicon, whereinnearest-neighbor coupled registers of spin qubits can be interfaced withother sparsely distributed registers via microwave photons. Furtherimprovements in cavity quality factors and spin-photon coupling rates,will allow additional time-domain demonstrations of cavity-assistedspin-spin coupling.

The system can be modeled with an effective Jaynes-Cummings Hamiltonian

${H = {{{\sum\limits_{j}{\frac{\hslash\omega_{j}}{2}\sigma_{j}^{z}}} + \hslash}❘{{\omega_{c}a^{\dagger}a} + {\hslash{g_{s,j}\left( {{\sigma_{j}^{+}a} + {\sigma_{j}^{-}a^{\dagger}}} \right)}}}}},$where j=L, R, and ℏω_(j) is the energy level splitting of the effectivej-spin, σ_(n) ^(μ) an are the Pauli operators in the j-spin Hilbertspace, ω_(c)/2π the cavity frequency and g_(s,j) is the effectivespin-cavity coupling rate. Dephasing can be introduced in our system bycoupling the spin-cavity system to external baths. Integrating out thesebaths in a Born-Markov approximation leads to a Lindblad master equationfor the spin-cavity dynamics

${\frac{d\rho}{dt} = {{- {\frac{i}{\hslash}\left\lbrack {H,\rho} \right\rbrack}} + {\sum\limits_{j}{\left( {{\gamma_{s,j}^{d}{\mathcal{D}\left\lbrack \sigma_{j}^{z} \right\rbrack}} + {\gamma_{s,j}^{r}{\mathcal{D}\left\lbrack \sigma_{j}^{-} \right\rbrack}}} \right)\rho}} + {\kappa{\mathcal{D}\lbrack\alpha\rbrack}\rho}}},$where

[A](ρ)=AρA^(†)−½(A^(†)Ap+ρA^(†)A) is the Lindblad super-operator,

_(s,j) ^(d) is the spin-dephasing rate,

_(s,j) ^(r) is the relaxation rate of the spins, and is the total cavitydecay rate. The spin linewidth (FWHM) is given by 2

_(s,j)=

_(s,j) ^(d)+

_(s,j) ^(r). Decompose κ=κ₁+κ₂+κ_(in) into the contributions from thedecay rates into the two transmission lines κ_(1,2) and an intrinsiccavity decay rate κ_(in).

The cavity transmission can be computed from the master equationevolution using input-output theory, which relates the output fieldoperator a_(n,out) of each transmission line to the input field operatora_(n,in) through the relationa _(n,out)(t)=√{square root over (κ_(n))}a(t)−a _(n,in)(t).where a(t) is found from the operator evolution under the Lindbladmaster equation with the addition of a drive termH_(drive)=Σ_(n)√{square root over (κ_(n))}(a^(†)a_(n,in)+h.c.). In thelimit of weak driving, √{square root over (κ_(n))}

a_(n)a_(in)

«κ, one can neglect the population of the excited states of theJaynes-Cummings model with more than one excitation, leaving the reducedspin-photon Hilbert space|0,↓,↓

,|0,↑,↓

,|0,↓,↑

,|1,↓,↓

where the first quantum number labels the number of photons in thecavity and the left/right spin labels refer to the L/R-spin directionquantized along the axis of the local magnetic field. Wheng_(s,j)Tr[ρa]«γ_(s,j), one can further neglect populations outside ofthe ground state |0,↓,↓

and only need to account for off-diagonal coherences in the masterequation evolution. In this limit, the complex transmission amplitudetakes the approximate form

$A = {\frac{\left\langle a_{2,{out}} \right\rangle}{\left\langle a_{1,{in}} \right\rangle} = {\frac{\sqrt{\kappa_{2}}T{r\left\lbrack {\rho a} \right\rbrack}}{\left\langle a_{1,{in}} \right\rangle} = \frac{- {i\left( {\sqrt{\kappa_{1}\kappa_{2}} + {\Delta_{0}/q}} \right)}}{{- \Delta_{0}} - {i\frac{\kappa}{2}} - {i{\sum_{\mathcal{j}}\frac{g_{s,{\mathcal{j}}}^{2}}{\gamma_{s,{\mathcal{j}}} + {i\delta_{\mathcal{j}}}}}}}}}$where Δ₀=ω_(d)−ω_(c) is the detuning of the cavity drive ω_(d) from thecavity frequency, δ_(j)=ω_(j)−ω_(d) is the detuning of the j-spin fromthe drive frequency, and a complex parameter q is introduced to accountfor Fano interference effects in the bare cavity transmission. Forsimplicity, all higher order corrections in g_(s,j)/q to the cavitytransmission are neglected.

From the spin-photon transitions in FIGS. 22C-D, the qubit frequency isfit and the magnetic susceptibility χ of the micromagnet is extracted,finding χ≈0.6 at ϕ=0° over small ranges of magnetic field. To generatethe fit in FIG. 23C, the angular dependence of the resonance frequenciesof the two DQDs is extracted using microwave spectroscopy in thedispersive regime. Substituting the extracted qubit frequencies ω_(j)into Eq. 5, the prediction for A/A₀ can be generated.

The fits to the R-spin and L-spin line cuts shown in FIGS. 24C-D aremodeled with Eq. 5, using the input parameters in Table I. The parameterg_(s) can remain as a free parameter in nonlinear regression fits andfix the remaining parameters extracted from separate measurements. Theerrors on g_(s) given in the main text are determined from these fits.The gradient field can be assumed between the two dots is ΔB_(x)=30 mTbased on previous experiments. t_(c,L/R) is the interdot tunnel couplingof the left (right) DQDs. To fit the data in FIGS. 24C-D with both spinson resonance, an effective model is used where the splitting of the twobright states are extracted using a single collective coupling g_(s,LR)and a single effective decay rate γ_(s,LR).

The present disclosure may comprise at least the following aspects.

Aspect 1. A device comprising, consisting of, or consisting essentiallyof: at least one semiconducting layer; one or more conducting layersconfigured to define at least two quantum states in the at least onesemiconducting layer and confine an electron in or more of the at leasttwo quantum states; a magnetic field source configured to generate aninhomogeneous magnetic field, wherein the inhomogeneous magnetic fieldcauses a first coupling of an electric charge state of the electron anda spin state of the electron; and a resonator configured to confine aphoton, wherein an electric-dipole interaction causes a second couplingof an electric charge state of the electron to an electric field of thephoton.

Aspect 2. The device of Aspect 1, wherein the photon is coupled to thespin state of the electron based on the first coupling and the secondcoupling.

Aspect 3. The device of any one of Aspects 1-2, wherein adjusting one ormore of a field strength or an angle of an external magnetic fieldapplied to one or more of the resonator or the at least onesemiconducting layer causes coupling of the spin state to the photon.

Aspect 4. The device of any one of Aspects 1-3, wherein causingresonance of the energy of the electron and the energy of the photoncauses coupling of the spin state to the photon.

Aspect 5. The device of any one of Aspects 1-4, wherein the firstcoupling comprises hybridization of the electric charge state of theelectron and the spin state of the electron.

Aspect 6. The device of any one of Aspects 1-5, wherein the secondcoupling comprises hybridization of the electric charge state of theelectron and the electric field of the photon.

Aspect 7. The device of any one of Aspects 1-6, wherein a spin-photoncoupling between the photon and the electron occurs with a spin-photoncoupling rate in a range of one or more of: about 10 Mhz to about 15 Mhzor about 5 MHz to about 150 MHz.

Aspect 8. The device of any one of Aspects 1-7, wherein the resonator isconfigured to couple the spin state of the electron to an additionalspin state of another electron separated from the electron by a distancein a range of one or more of: about 1 mm to about 5 mm, about 2 mm toabout 4 mm, about 3 mm to about 4 mm, about 3 mm to about 4 mm, about 1mm to about 10 mm, or about 3 mm to about 8 mm.

Aspect 9. The device of any one of Aspects 1-8, wherein the resonator iscoupled to a structure configured to confine an additional electron,wherein the structure is in a material stack separate from a materialstack comprising the electron, wherein the photon mediates coupling of aspin state of the electron and a spin state of the additional electron.

Aspect 10. The device of Aspect 9, wherein adjusting one or more of anangle or an amplitude of an external magnetic field allows forsimultaneous tuning of the spin state of the electron and the spin stateof the additional electron in resonance with one or more of the photonor the resonator.

Aspect 11. The device of any one of Aspects 1-10, wherein the deviceenables coupling of a plurality of quantum states to the cavity togenerate long range quantum gates between the plurality of quantumstates.

Aspect 12. The device of any one of Aspects 1-11, wherein the at leastone semiconducting layer comprises one or more of silicon, germanium, orsilicon germanium.

Aspect 13. The device of any one of Aspects 1-12, wherein the at leastone semiconducting layer comprises an isotopically enriched material.

Aspect 14. The device of Aspect 13, wherein the isotopically enrichedmaterial comprises isotopically enriched silicon.

Aspect 15. The device of Aspect 14, wherein the isotopically enrichedsilicon comprises nuclear spin zero silicon 28 isotope.

Aspect 16. The device of any one of Aspects 1-15, wherein the at leastone semiconducting layer comprises a silicon layer disposed on a silicongermanium layer.

Aspect 17. The device of any one of Aspects 1-16, wherein the one ormore conducting layers comprises a layer electrically coupled to theresonator.

Aspect 18. The device of Aspect 17, wherein the layer electricallycoupled to the resonator comprises a split-gate layer comprising a firstgate and a second gate separated from the first gate by a gap.

Aspect 19. The device of Aspect 18, wherein one or more of a size or alocation of the gap increases (e.g., maximizes, optimizes) an electricfield in a region of the at least two quantum state.

Aspect 20. The device of any one of Aspects 18-19, wherein one or moreof a size or a location of the gap increases (e.g., maximizes,optimizes) coherent coupling between one or more of the at least twoquantum states and the photon.

Aspect 21. The device of any one of Aspects 18-20, wherein the gapoccurs in a layer of a material stack comprising the one or moreconducting layers, and wherein the gap is below or above one or more ofa plunger gate or a barrier gate.

Aspect 22. The device of any one of Aspects 1-21, wherein the at leasttwo quantum states comprise a gate defined silicon double quantum dot.

Aspect 23. The device of any one of Aspects 1-22, wherein the one ormore conducting layers comprise a first conducting layer comprising oneor more barrier gates configured to define the at least two quantumstates, and wherein one or more conducting layers comprise a secondconducting layer comprising one or more plunger gates configured tocause the first electron to move between the at least two quantumstates.

Aspect 24. The device of Aspect 23, wherein a first plunger gate of theone or more plunger gates is electrically coupled to the resonator.

Aspect 25. The device of any one of Aspects 1-24, wherein the at leastone magnetic field source comprises at least one micro-magnet disposedin a material stack comprising the at least one semiconducting layer.

Aspect 26. The device of any one of Aspects 1-25, wherein the at leastone magnetic field source comprises a first micro-magnet and a secondmicro-magnet separated from the first micro-magnet.

Aspect 27. The device of any one of Aspects 1-26, wherein the at leastone magnetic field source is tilted such that a long axis of the atleast one magnetic field source is angled relative to an axis betweenthe at least two quantum states.

Aspect 28. The device of any one of Aspects 1-27, wherein the at leastone magnetic field source comprises a current carrying wire configuredto generate a magnetic field to enable spin-photon coupling.

Aspect 29. The device of any one of Aspects 1-8, wherein the resonatorcomprises one or more of a cavity, a superconductive cavity, or amicrowave cavity.

Aspect 30. The device of any one of Aspects 1-29, wherein resonatorcomprises a path, a first mirror disposed on the path, and a secondmirror disposed on the path opposite the first mirror.

Aspect 31. The device of any one of Aspects 1-30, wherein the resonatorcomprises a center pin adjacent a vacuum gap.

Aspect 32. The device of Aspect 31, wherein the center pin has athickness in a range of one or more of: about 50 nm to about 10 μm. orabout 0.5 μm to about 1 μm

Aspect 33. The device of any one of Aspects 31-32, wherein the vacuumgap has a thickness in a range of one or more of: about 15 μm to about25 μm or about 50 nm to about 30 μm.

Aspect 34. The device of any one of Aspects 1-33, wherein the resonatorhas an impedance in a range of one or more of: about 50Ω to about 200Ω,about 50Ω to about 300Ω, about 50Ω to about 3 kΩ, about 200Ω to about300Ω, about 200Ω to about 200Ω, about 1 kΩ to about 2 kΩ, about 200Ω toabout 3 kΩ, or about 1 kΩ to about 3kΩ.

Aspect 35. The device of any one of Aspects 1-34, wherein a length ofthe resonator is between about 10,000 and about 1,000,000 times a lengthin which the electron is confined.

Aspect 36. The device of any one of Aspects 1-35, wherein the resonatorhas a center frequency in a range of one or more of: about 2 GHz toabout 12 GHz or about 7 GHz and about 8 GHz.

Aspect 37. A system comprising, consisting of, or consisting essentiallyof: a first structure configured to define at least two first quantumstates in a first semiconducting layer and confine a first electron inor more of the at least two first quantum states, wherein the firststructure comprises at least one magnetic field source configured tosupply an inhomogeneous magnetic field to the first electron; a secondstructure configured to define at least two second quantum states in asecond semiconducting layer and confine a second electron in or more ofthe at least two second quantum states; and a resonator disposedadjacent the first structure and the second structure, wherein tuning ofan external magnetic field allows for a photon in the resonator tomediate coupling a first spin state of the first electron to a secondspin state of the second electron.

Aspect 38. The system of Aspect 37, wherein the inhomogeneous magneticfield enables a first coupling of an electric charge state of the firstelectron and a first spin state of the first electron, and wherein anelectric-dipole interaction causes a second coupling of an electriccharge state of the first electron to an electric field of the photon.

Aspect 39. The system of Aspect 38, wherein the photon is coupled to thefirst spin state of the first electron based on the first coupling andthe second coupling.

Aspect 40. The system of any one of Aspects 38-39, wherein the firstcoupling comprises hybridization of the electric charge state of thefirst electron and the first spin state of the first electron.

Aspect 41. The system of any one of Aspects 38-40, wherein the secondcoupling comprises hybridization of the electric charge state of thefirst electron and the electric field of the photon.

Aspect 42. The system of any one of Aspects 37-41, wherein adjusting oneor more of a field strength or an angle of an external magnetic fieldapplied to one or more of the resonator or the first structure causescoupling of the first spin state to the photon.

Aspect 43. The system of any one of Aspects 37-42, wherein causingresonance of an energy of the first electron and an energy of the photoncauses coupling of the first spin state to the photon.

Aspect 44. The system of any one of Aspects 37-43, wherein a spin-photoncoupling between the photon and the first electron occurs with aspin-photon coupling rate in a range of one or more of: about 10 Mhz toabout 15 Mhz or about 5 MHz to about 150 MHz.

Aspect 45. The system of any one of Aspects 37-44, wherein the firstelectron is separated from the second electron by a distance in a rangeof one or more of: about 1 mm to about 5 mm, about 2 mm to about 4 mm,about 3 mm to about 4 mm, about 3 mm to about 4 mm, about 1 mm to about10 mm, or about 3 mm to about 8 mm.

Aspect 46. The system of any one of Aspects 37-45, wherein the firststructure is comprised in a material stack separate from a materialstack comprising the second structure.

Aspect 47. The system of Aspect 46, wherein adjusting one or more of anangle or an amplitude of an external magnetic field allows forsimultaneous tuning of the first spin state of the first electron andthe second spin state of the second electron in resonance with one ormore of the photon or the resonator.

Aspect 48. The system of any one of Aspects 37-47, wherein the systemenables coupling of a plurality of quantum states to the cavity togenerate long range quantum gates between the plurality of quantumstates.

Aspect 49. The system of any one of Aspects 37-48, wherein one or moreof the first semiconducting layer or the second semiconducting layercomprises one or more of silicon, germanium, or silicon germanium.

Aspect 50. The system of any one of Aspects 37-49, wherein one or moreof the first semiconducting layer or the second semiconducting layercomprises an isotopically enriched material.

Aspect 51. The system of Aspect 50, wherein the isotopically enrichedmaterial comprises isotopically enriched silicon.

Aspect 52. The system of Aspect 51, wherein the isotopically enrichedsilicon comprises nuclear spin zero silicon 28 isotope.

Aspect 53. The system of any one of Aspects 37-52, wherein one or moreof the first semiconducting layer or the second semiconducting layercomprises a silicon layer disposed on a silicon germanium layer.

Aspect 54. The system of any one of Aspects 37-53, wherein one or moreof the first semiconducting layer or the second semiconducting layercomprises a layer electrically coupled to the resonator.

Aspect 55. The system of Aspect 54, wherein the layer electricallycoupled to the resonator comprises a split-gate layer comprising a firstgate and a second gate separated from the first gate by a gap.

Aspect 56. The system of Aspect 55, wherein one or more of a size or alocation of the gap increases (e.g., maximizes, optimizes) an electricfield in a region of the at least two first quantum states.

Aspect 57. The system of any one of Aspects 55-56, wherein one or moreof a size or the location of the gap increases (e.g., maximizes,optimizes) coherent coupling between one or more of the at least twofirst quantum states and the photon.

Aspect 58. The system of any one of Aspects 55-57, wherein the gapoccurs in a layer of a material stack comprising one or more conductinglayers, and wherein the gap is below or above one or more of a plungergate or a barrier gate.

Aspect 59. The system of any one of Aspects 37-58, wherein the at leasttwo first quantum states comprise a gate defined silicon double quantumdot.

Aspect 60. The system of any one of Aspects 37-59, wherein one or moreof the first structure or the second structure comprise one or moreconducting layers, wherein the one or more conducting layers comprise afirst conducting layer comprising one or more barrier gates configuredto define the at least two first quantum states, and wherein the one ormore conducting layers comprise a second conducting layer comprising oneor more plunger gates configured to cause the first electron to movebetween the at least two first quantum states.

Aspect 61. The system of Aspect 60, wherein a first plunger gate of theone or more plunger gates is electrically coupled to the resonator.

Aspect 62. The system of any one of Aspects 37-61, wherein the at leastone magnetic field source comprises at least one micro-magnet disposedin a material stack comprising the first semiconducting layer.

Aspect 63. The system of any one of Aspects 37-62, wherein the at leastone magnetic field source comprises a first micro-magnet and a secondmicro-magnet separated from the first micro-magnet.

Aspect 64. The system of any one of Aspects 37-63, wherein the at leastone magnetic field source is tilted such that a long axis of the atleast one magnetic field source is angled relative to an axis betweenthe at least two first quantum states of the first structure.

Aspect 65. The system of any one of Aspects 37-64, wherein the at leastone magnetic field source comprises a current carrying wire configuredto generate a magnetic field to enable spin-photon coupling.

Aspect 66. The system of any one of Aspects 37-65, wherein the resonatorcomprises one or more of a cavity, a superconductive cavity, or amicrowave cavity.

Aspect 67. The system of any one of Aspects 37-66, wherein resonatorcomprises a path, a first minor disposed on the path, and a secondmirror disposed on the path opposite the first mirror.

Aspect 68. The system of any one of Aspects 37-67, wherein the resonatorcomprises a center pin adjacent a vacuum gap.

Aspect 69. The system of Aspect 68, wherein the center pin has athickness in a range of one or more of: about 50 nm to about 10 μm. orabout 0.5 μm to about 1 μm

Aspect 70. The system of any one of Aspects 68-69, wherein the vacuumgap has a thickness in a range of one or more of: about 15 μm to about25 μm or about 50 nm to about 30 μm.

Aspect 71. The system of any one of Aspects 37-70, wherein the resonatorhas an impedance in a range of one or more of: about 50Ω to about 200Ω,about 50Ω to about 300Ω, about 50Ω to about 3 kΩ, about 200Ω to about300Ω, about 200Ω to about 2 kΩ, about 1 kΩ to about 2 kΩ, about 200Ω toabout 3 kΩ, or about 1 kΩ to about 3 kΩ.

Aspect 72. The system of any one of Aspects 37-71, wherein a length ofthe resonator is between about 10,000 and about 1,000,000 times a lengthin which the first electron is confined.

Aspect 73. The system of any one of Aspects 37-72, wherein the resonatorhas a center frequency in a range of one or more of: about 2 GHz toabout 12 GHz or about 7 GHz and about 8 GHz.

Aspect 74. A method comprising, consisting of, or consisting essentiallyof: confining a first electron with a first structure, wherein the firststructure defines two quantum states in at least one semiconductinglayer using one or more conducting layers; causing, based on aninhomogeneous magnetic field source, a first coupling of an electriccharge state of the first electron and a first spin state of the firstelectron; causing, based on an electric-dipole interaction, a secondcoupling of an electric field of a photon in a resonator to an orbitalstate of the first electron; and causing, based on the first couplingand the second coupling, a coupling of the photon and the first spinstate of the first electron.

Aspect 75. The method of Aspect 74, wherein causing the coupling of thephoton and the first spin state of the first electron comprisesadjusting one or more of a field strength or an angle of an externalmagnetic field applied to one or more of the resonator or the at leastone semiconducting layer to cause coupling of the first spin state tothe photon.

Aspect 76. The method of any one of Aspects 74-75, wherein causing thesecond coupling comprises causing resonance of an energy of the firstelectron and an energy of the photon.

Aspect 77. The method of any one of Aspects 74-76, wherein the firstcoupling comprises hybridization of an electric charge state of thefirst electron and the first spin state of the first electron.

Aspect 78. The method of any one of Aspects 74-77, wherein the secondcoupling comprises hybridization of an electric charge state of thefirst electron and the electric field of the photon.

Aspect 79. The method of any one of Aspects 74-78, wherein a spin-photoncoupling between the photon and the first electron occurs with aspin-photon coupling rate in a range of one or more of: about 10 Mhz toabout 15 Mhz or about 5 MHz to about 150 MHz.

Aspect 80. The method of any one of Aspects 74-79, further comprisingcausing coupling of the first spin state of the first electron to asecond spin state of a second electron.

Aspect 81. The method of Aspect 80, wherein the first electron isseparated from the second electron by a distance in a range of one ormore of: about 1 mm to about 5 mm, about 2 mm to about 4 mm, about 3 mmto about 4 mm, about 3 mm to about 4 mm, about 1 mm to about 10 mm, orabout 3 mm to about 8 mm.

Aspect 82. The method of any one of Aspects 80-81, wherein the resonatoris coupled to a structure configured to confine the second electron,wherein the structure is in a material stack separate from a materialstack comprising the first electron, wherein the photon mediatescoupling of a first spin state of the first electron and a second spinstate of the second electron.

Aspect 83. The method of Aspect 82, wherein causing coupling of thefirst spin state and the second spin state comprises adjusting one ormore of an angle or an amplitude of an external magnetic field to allowsimultaneously tuning of the first spin state of the first electron andthe second spin state of the second electron in resonance with one ormore of the photon or the resonator.

Aspect 84. The method of any one of Aspects 74-83, further coupling of aplurality of quantum states to the resonator to generate long rangequantum gates between the plurality of quantum states.

Aspect 85. The method of any one of Aspects 74-84, wherein the at leastone semiconducting layer comprises one or more of silicon, germanium, orsilicon germanium.

Aspect 86. The method of any one of Aspects 74-85, wherein the at leastone semiconducting layer comprises an isotopically enriched material.

Aspect 87. The method of Aspect 86, wherein the isotopically enrichedmaterial comprises isotopically enriched silicon.

Aspect 88. The method of Aspect 87, wherein the isotopically enrichedsilicon comprises nuclear spin zero silicon 28 isotope.

Aspect 89. The method of any one of Aspects 74-88, wherein the at leastone semiconducting layer comprises a silicon layer disposed on a silicongermanium layer.

Aspect 90. The method of any one of Aspects 74-89, wherein the one ormore conducting layers comprise a layer electrically coupled to theresonator.

Aspect 91. The method of Aspect 90, wherein the layer electricallycoupled to the resonator comprises a split-gate layer comprising a firstgate and a second gate separated from the first gate by a gap.

Aspect 92. The method of Aspect 91, wherein one or more of a size or alocation of the gap increases (e.g., maximizes, optimizes) an electricfield in a region of the at least two quantum state.

Aspect 93. The method of any one of Aspects 91-92, wherein one or moreof a size or a location of the gap increases (e.g., maximizes,optimizes) coherent coupling between one or more of the at least twoquantum states and the photon.

Aspect 94. The method of any one of Aspects 91-93, wherein the gapoccurs in a layer of a material stack comprising the one or moreconducting layers, and wherein the gap is below or above one or more ofa plunger gate or a barrier gate.

Aspect 95. The method of any one of Aspects 74-94, wherein the at leasttwo quantum states comprise a gate defined silicon double quantum dot.

Aspect 96. The method of any one of Aspects 74-95, wherein the one ormore conducting layers comprise a first conducting layer comprising oneor more barrier gates configured to define the at least two quantumstates, and wherein one or more conducting layers comprise a secondconducting layer comprising one or more plunger gates configured tocause the first electron to move between the at least two quantumstates.

Aspect 97. The method of Aspect 96, wherein a first plunger gate of theone or more plunger gates is electrically coupled to the resonator.

Aspect 98. The method of any one of Aspects 74-97, wherein theinhomogeneous magnetic field source comprises at least one micro-magnetdisposed in a material stack comprising the at least one semiconductinglayer.

Aspect 99. The method of any one of Aspects 74-98, wherein theinhomogeneous magnetic field source comprises a first micro-magnet and asecond micro-magnet separated from the first micro-magnet.

Aspect 100. The method of any one of Aspects 74-99, wherein theinhomogeneous magnetic field source is tilted such that a long axis ofthe at least one magnetic field source is angled relative to an axisbetween the at least two quantum states.

Aspect 101. The method of any one of Aspects 74-100, wherein theinhomogeneous magnetic field source comprises a current carrying wireconfigured to generate a magnetic field to enable spin-photon coupling.

Aspect 102. The method of any one of Aspects 74-101, wherein theresonator comprises one or more of a cavity, a superconductive cavity,or a microwave cavity.

Aspect 103. The method of any one of Aspects 74-102, wherein resonatorcomprises a path, a first mirror disposed on the path, and a secondmirror disposed on the path opposite the first mirror.

Aspect 104. The method of any one of Aspects 74-103, wherein theresonator comprises a center pin adjacent a vacuum gap.

Aspect 105. The method of Aspect 104, wherein the center pin has athickness in a range of one or more of: about 50 nm to about 10 μm. orabout 0.5 μm to about 1 μm.

Aspect 106. The method of any one of Aspects 104-105, wherein the vacuumgap has a thickness in a range of one or more of: about 15 μm to about25 μm or about 50 nm to about 30 μm.

Aspect 107. The method of any one of Aspects 74-106, wherein theresonator has an impedance in a range of one or more of: about 50Ω toabout 200Ω, about 50Ω to about 300Ω, about 50Ω to about 3 kΩ, about 200Ωto about 300Ω, about 200Ω to about 2 kΩ, about 1 kΩ to about 2 kΩ, about200Ω to about 3 kΩ, or about 1 kΩ to about 3 kΩ.

Aspect 108. The method of any one of Aspects 74-107, wherein a length ofthe resonator is between about 10,000 and about 1,000,000 times a lengthin which one or more of the first electron or second electron isconfined.

Aspect 109. The method of any one of Aspects 74-108, wherein theresonator has a center frequency in a range of one or more of: about 2GHz to about 12 GHz or about 7 GHz and about 8 GHz.

Aspect 110. A spin to photon transducer, comprising, consisting of, orconsisting essentially of: a semiconductor heterostructure adapted toconfine electrons in a 2-dimensional plane; a plurality of overlappingconducting layers of electrodes, configured to confine an electronwithin a double quantum dot; a superconducting cavity positionedadjacent to the double quantum dot; and a micro-magnet positioned overthe double quantum dot.

Aspect 111. The spin to photon transducer of Aspect 110, wherein thesemiconductor heterostructure is Si/SiGe.

Aspect 112. The spin to photon transducer of any one of Aspects 110-111,wherein the electrodes are aluminum.

Aspect 113. The spin to photon transducer of any one of Aspects 110-112,wherein the superconducting cavity is niobium.

Aspect 114. The spin to photon transducer of any one of Aspects 110-113,wherein the superconducting cavity has a center frequency between 7 and8 GHz.

Aspect 115. The spin to photon transducer of any one of Aspects 110-114,wherein a length of the superconducting cavity is between 10,000 and1,000,000 times a length in which the single electron is confined.

Aspect 116. The spin to photon transducer of any one of Aspects 110-115,wherein the plurality of overlapping conducting layers consists of threelayers including a first layer consisting of at least 2 gates, a secondlayer consisting of at least 2 plunger gates, and third layer consistingof at least 3 barrier gates.

Aspect 117. A method for coupling the electric field of the cavityphoton to the spin state of the electron, the method comprising,consisting of, or consisting essentially of the steps of: confiningelectrons in a 2-dimensional plane within a semiconductorheterostructure; confining an electron within a double quantum dot;coupling an electric field of a superconducting cavity photon to anorbital state of the electron through electric dipole interaction; andproducing a local magnetic field gradient to hybridize the orbital stateof the electron with its spin state.

Aspect 118. The method of Aspect 117, wherein the semiconductorheterostructure is Si/SiGe.

Aspect 119. The method of any one of Aspects 117-118, wherein thesuperconducting cavity is niobium.

Aspect 120. The method of any one of Aspects 117-119, further comprisingtuning the transition frequency of the electron spin to about the cavityfrequency.

It is to be understood that the methods and systems are not limited tospecific methods, specific components, or to particular implementations.It is also to be understood that the terminology used herein is for thepurpose of describing particular embodiments only and is not intended tobe limiting.

As used in the specification and the appended claims, the singular forms“a,” “an,” and “the” include plural referents unless the context clearlydictates otherwise. Ranges may be expressed herein as from “about” oneparticular value, and/or to “about” another particular value. When sucha range is expressed, another embodiment includes from the oneparticular value and/or to the other particular value. Similarly, whenvalues are expressed as approximations, by use of the antecedent“about,” it will be understood that the particular value forms anotherembodiment. It will be further understood that the endpoints of each ofthe ranges are significant both in relation to the other endpoint, andindependently of the other endpoint.

“Optional” or “optionally” means that the subsequently described eventor circumstance may or may not occur, and that the description includesinstances where said event or circumstance occurs and instances where itdoes not.

Throughout the description and claims of this specification, the word“comprise” and variations of the word, such as “comprising” and“comprises,” means “including but not limited to,” and is not intendedto exclude, for example, other components, integers or steps.“Exemplary” means “an example of” and is not intended to convey anindication of a preferred or ideal embodiment. “Such as” is not used ina restrictive sense, but for explanatory purposes.

Components are described that may be used to perform the describedmethods and systems. When combinations, subsets, interactions, groups,etc., of these components are described, it is understood that whilespecific references to each of the various individual and collectivecombinations and permutations of these may not be explicitly described,each is specifically contemplated and described herein, for all methodsand systems. This applies to all aspects of this application including,but not limited to, operations in described methods. Thus, if there area variety of additional operations that may be performed it isunderstood that each of these additional operations may be performedwith any specific embodiment or combination of embodiments of thedescribed methods.

As will be appreciated by one skilled in the art, the methods andsystems may take the form of an entirely hardware embodiment, anentirely software embodiment, or an embodiment combining software andhardware aspects. Furthermore, the methods and systems may take the formof a computer program product on a computer-readable storage mediumhaving computer-readable program instructions (e.g., computer software)embodied in the storage medium. More particularly, the present methodsand systems may take the form of web-implemented computer software. Anysuitable computer-readable storage medium may be utilized including harddisks, CD-ROMs, optical storage devices, or magnetic storage devices.

Embodiments of the methods and systems are described herein withreference to block diagrams and flowchart illustrations of methods,systems, apparatuses and computer program products. It will beunderstood that each block of the block diagrams and flowchartillustrations, and combinations of blocks in the block diagrams andflowchart illustrations, respectively, may be implemented by computerprogram instructions. These computer program instructions may be loadedon a general-purpose computer, special-purpose computer, or otherprogrammable data processing apparatus to produce a machine, such thatthe instructions which execute on the computer or other programmabledata processing apparatus create a means for implementing the functionsspecified in the flowchart block or blocks.

These computer program instructions may also be stored in acomputer-readable memory that may direct a computer or otherprogrammable data processing apparatus to function in a particularmanner, such that the instructions stored in the computer-readablememory produce an article of manufacture including computer-readableinstructions for implementing the function specified in the flowchartblock or blocks. The computer program instructions may also be loadedonto a computer or other programmable data processing apparatus to causea series of operational steps to be performed on the computer or otherprogrammable apparatus to produce a computer-implemented process suchthat the instructions that execute on the computer or other programmableapparatus provide steps for implementing the functions specified in theflowchart block or blocks.

The various features and processes described above may be usedindependently of one another, or may be combined in various ways. Allpossible combinations and sub-combinations are intended to fall withinthe scope of this disclosure. In addition, certain methods or processblocks may be omitted in some implementations. The methods and processesdescribed herein are also not limited to any particular sequence, andthe blocks or states relating thereto may be performed in othersequences that are appropriate. For example, described blocks or statesmay be performed in an order other than that specifically described, ormultiple blocks or states may be combined in a single block or state.The example blocks or states may be performed in serial, in parallel, orin some other manner. Blocks or states may be added to or removed fromthe described example embodiments. The example systems and componentsdescribed herein may be configured differently than described. Forexample, elements may be added to, removed from, or rearranged comparedto the described example embodiments.

While the methods and systems have been described in connection withpreferred embodiments and specific examples, it is not intended that thescope be limited to the particular embodiments set forth, as theembodiments herein are intended in all respects to be illustrativerather than restrictive.

It will be apparent to those skilled in the art that variousmodifications and variations may be made without departing from thescope or spirit of the present disclosure. Other embodiments will beapparent to those skilled in the art from consideration of thespecification and practices described herein. It is intended that thespecification and example figures be considered as exemplary only, witha true scope and spirit being indicated by the following claims.

What is claimed:
 1. A method comprising: confining a first electron with a first structure, wherein the first structure defines at least two quantum states in at least one semiconducting layer using one or more conducting layers; causing, based on an inhomogeneous magnetic field source, a first coupling of an electric charge state of the first electron and a first spin state of the first electron; causing, based on an electric-dipole interaction, a second coupling of an electric field of a photon in a resonator to an orbital state of the first electron; and causing, based on the first coupling and the second coupling, a coupling of the photon and the first spin state of the first electron.
 2. The method of claim 1, wherein causing the coupling of the photon and the first spin state of the first electron comprises adjusting one or more of a field strength or an angle of an external magnetic field applied to one or more of the resonator or the at least one semiconducting layer to cause coupling of the first spin state to the photon.
 3. The method of claim 1, wherein causing the second coupling comprises causing resonance of an energy of the first electron and an energy of the photon.
 4. The method of claim 1, wherein the first coupling comprises hybridization of an electric charge state of the first electron and the first spin state of the first electron.
 5. The method of claim 1, wherein the second coupling comprises hybridization of an electric charge state of the first electron and the electric field of the photon.
 6. The method of claim 1, further comprising causing coupling of the first spin state of the first electron to a second spin state of a second electron.
 7. The method of claim 6, wherein the resonator is coupled to a structure configured to confine the second electron, wherein the structure is in a material stack separate from a material stack comprising the first electron, wherein the photon mediates coupling of the first spin state of the first electron and a second spin state of the second electron.
 8. The method of claim 7, wherein causing coupling of the first spin state and the second spin state comprises adjusting one or more of an angle or an amplitude of an external magnetic field to allow simultaneously tuning of the first spin state of the first electron and the second spin state of the second electron in resonance with one or more of the photon or the resonator.
 9. The method of claim 1, further comprising coupling of a plurality of quantum states to the resonator to generate long range quantum gates between the plurality of quantum states.
 10. The method of claim 1, wherein the at least one semiconducting layer comprises an isotopically enriched material.
 11. The method of claim 1, wherein the one or more conducting layers comprise a layer electrically coupled to the resonator.
 12. The method of claim 11, wherein the layer electrically coupled to the resonator comprises a split-gate layer comprising a first gate and a second gate separated from the first gate by a gap.
 13. The method of claim 1, wherein the at least two quantum states comprise a gate defined silicon double quantum dot.
 14. The method of claim 1, wherein the one or more conducting layers comprise a first conducting layer comprising one or more barrier gates configured to define the at least two quantum states, and wherein one or more conducting layers comprise a second conducting layer comprising one or more plunger gates configured to cause the first electron to move between the at least two quantum states.
 15. The method of claim 14, wherein a first plunger gate of the one or more plunger gates is electrically coupled to the resonator.
 16. The method of claim 1, wherein the inhomogeneous magnetic field source comprises at least one micro-magnet disposed in a material stack comprising the at least one semiconducting layer.
 17. The method of claim 1, wherein the inhomogeneous magnetic field source comprises a first micro-magnet and a second micro-magnet separated from the first micro-magnet.
 18. The method of claim 1, wherein the inhomogeneous magnetic field source is tilted such that a long axis of the inhomogeneous magnetic field source is angled relative to an axis between the at least two quantum states.
 19. The method of claim 1, wherein the inhomogeneous magnetic field source comprises a current carrying wire configured to generate a magnetic field to enable spin-photon coupling.
 20. The method of claim 1, wherein the resonator comprises one or more of a cavity, a superconductive cavity, or a microwave cavity. 